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Least common denominator cos(a)/2*sin(2)-1-sin(a)/2*cos(2)-1

An expression to simplify:

The solution

You have entered [src]
cos(a)              sin(a)           
------*sin(2) - 1 - ------*cos(2) - 1
  2                   2              
$$\left(- \frac{\sin{\left(a \right)}}{2} \cos{\left(2 \right)} + \left(\frac{\cos{\left(a \right)}}{2} \sin{\left(2 \right)} - 1\right)\right) - 1$$
(cos(a)/2)*sin(2) - 1 - sin(a)/2*cos(2) - 1
General simplification [src]
     sin(-2 + a)
-2 - -----------
          2     
$$- \frac{\sin{\left(a - 2 \right)}}{2} - 2$$
-2 - sin(-2 + a)/2
Combining rational expressions [src]
-4 + cos(a)*sin(2) - cos(2)*sin(a)
----------------------------------
                2                 
$$\frac{- \sin{\left(a \right)} \cos{\left(2 \right)} + \sin{\left(2 \right)} \cos{\left(a \right)} - 4}{2}$$
(-4 + cos(a)*sin(2) - cos(2)*sin(a))/2
Rational denominator [src]
-4 + cos(a)*sin(2) - cos(2)*sin(a)
----------------------------------
                2                 
$$\frac{- \sin{\left(a \right)} \cos{\left(2 \right)} + \sin{\left(2 \right)} \cos{\left(a \right)} - 4}{2}$$
(-4 + cos(a)*sin(2) - cos(2)*sin(a))/2
Common denominator [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
           2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
Numerical answer [src]
-2.0 + 0.454648713412841*cos(a) + 0.208073418273571*sin(a)
-2.0 + 0.454648713412841*cos(a) + 0.208073418273571*sin(a)
Combinatorics [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
           2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
Powers [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
           2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
                        / I*a    -I*a\     / -2*I    2*I\                 
       /   -2*I    2*I\ |e      e    |     |e       e   | /   -I*a    I*a\
     I*\- e     + e   /*|---- + -----|   I*|----- + ----|*\- e     + e   /
                        \ 4       4  /     \  2      2  /                 
-2 - --------------------------------- + ---------------------------------
                     2                                   4                
$$- \frac{i \left(e^{2 i} - e^{- 2 i}\right) \left(\frac{e^{i a}}{4} + \frac{e^{- i a}}{4}\right)}{2} + \frac{i \left(\frac{e^{- 2 i}}{2} + \frac{e^{2 i}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{4} - 2$$
-2 - i*(-exp(-2*i) + exp(2*i))*(exp(i*a)/4 + exp(-i*a)/4)/2 + i*(exp(-2*i)/2 + exp(2*i)/2)*(-exp(-i*a) + exp(i*a))/4
Assemble expression [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
           2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
Expand expression [src]
     sin(a)      2                                 
-2 + ------ - cos (1)*sin(a) + cos(1)*cos(a)*sin(1)
       2                                           
$$- \sin{\left(a \right)} \cos^{2}{\left(1 \right)} + \frac{\sin{\left(a \right)}}{2} + \sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(a \right)} - 2$$
-2 + sin(a)/2 - cos(1)^2*sin(a) + cos(1)*cos(a)*sin(1)
Trigonometric part [src]
          /     a\   
       tan|-1 + -|   
          \     2/   
-2 - ----------------
            2/     a\
     1 + tan |-1 + -|
             \     2/
$$-2 - \frac{\tan{\left(\frac{a}{2} - 1 \right)}}{\tan^{2}{\left(\frac{a}{2} - 1 \right)} + 1}$$
             1         
-2 - ------------------
          /         pi\
     2*sec|-2 + a - --|
          \         2 /
$$-2 - \frac{1}{2 \sec{\left(a - 2 - \frac{\pi}{2} \right)}}$$
          /     a\   
       cot|-1 + -|   
          \     2/   
-2 - ----------------
            2/     a\
     1 + cot |-1 + -|
             \     2/
$$-2 - \frac{\cot{\left(\frac{a}{2} - 1 \right)}}{\cot^{2}{\left(\frac{a}{2} - 1 \right)} + 1}$$
        /         pi\
     cos|-2 + a - --|
        \         2 /
-2 - ----------------
            2        
$$- \frac{\cos{\left(a - 2 - \frac{\pi}{2} \right)}}{2} - 2$$
     -sin(-2 + a) + sin(2 + a)      2/a\ /     1           2   \    /a\
-2 + ------------------------- - sin |-|*|----------- - sin (1)|*cot|-|
                 4                   \2/ |       2             |    \2/
                                         \1 + tan (1)          /       
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} \right)} - 2$$
     (-1 - cos(4) + 2*cos(2))*sin(a)     2*cos(a)*sin(2)  
-2 - ------------------------------- + -------------------
                                  2                   2   
       1 - cos(4) + 2*(1 - cos(2))          2      sin (2)
                                       4*sin (1) + -------
                                                      2   
                                                   sin (1)
$$- \frac{\left(-1 + 2 \cos{\left(2 \right)} - \cos{\left(4 \right)}\right) \sin{\left(a \right)}}{- \cos{\left(4 \right)} + 1 + 2 \left(1 - \cos{\left(2 \right)}\right)^{2}} + \frac{2 \sin{\left(2 \right)} \cos{\left(a \right)}}{\frac{\sin^{2}{\left(2 \right)}}{\sin^{2}{\left(1 \right)}} + 4 \sin^{2}{\left(1 \right)}} - 2$$
                   2                                                      
     /        2/a\\           2/a\                                        
     |-1 + cot |-||      4*cot |-|                                        
     \         \4//            \4/      /     1             1     \    /a\
     --------------- - --------------   |----------- - -----------|*cot|-|
                   2                2   |       1             2   |    \2/
      /       2/a\\    /       2/a\\    |1 + -------   1 + cot (1)|       
      |1 + cot |-||    |1 + cot |-||    |       2                 |       
      \        \4//    \        \4//    \    cot (1)              /       
-2 + -------------------------------- - ----------------------------------
           /       1   \                                  2/a\            
           |1 + -------|*cot(1)                    1 + cot |-|            
           |       2   |                                   \2/            
           \    cot (1)/                                                  
$$\frac{\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{4 \cot^{2}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}}{\left(1 + \frac{1}{\cot^{2}{\left(1 \right)}}\right) \cot{\left(1 \right)}} - 2 - \frac{\left(- \frac{1}{\cot^{2}{\left(1 \right)} + 1} + \frac{1}{1 + \frac{1}{\cot^{2}{\left(1 \right)}}}\right) \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}$$
     /             2                 \                                            
     |/       2/a\\           2/a\   |                                            
     ||1 - tan |-||      4*tan |-|   |                                            
     |\        \4//            \4/   |          /     1             1     \    /a\
     |-------------- - --------------|*tan(1)   |----------- - -----------|*tan|-|
     |             2                2|          |       2             1   |    \2/
     |/       2/a\\    /       2/a\\ |          |1 + tan (1)   1 + -------|       
     ||1 + tan |-||    |1 + tan |-|| |          |                     2   |       
     \\        \4//    \        \4// /          \                  tan (1)/       
-2 + ---------------------------------------- - ----------------------------------
                          2                                       2/a\            
                   1 + tan (1)                             1 + tan |-|            
                                                                   \2/            
$$\frac{\left(\frac{\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} - 2 - \frac{\left(- \frac{1}{\frac{1}{\tan^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
            1                 1       
-2 + --------------- - ---------------
     2*csc(2)*sec(a)   2*csc(a)*sec(2)
$$-2 + \frac{1}{2 \csc{\left(2 \right)} \sec{\left(a \right)}} - \frac{1}{2 \csc{\left(a \right)} \sec{\left(2 \right)}}$$
     /     1             1     \                                    
     |----------- - -----------|*sin(a)                             
     |       2             1   |                                    
     |1 + tan (1)   1 + -------|          /   2/a\      2/a\\       
     |                     2   |          |cos |-| - sin |-||*tan(1)
     \                  tan (1)/          \    \2/       \2//       
-2 - ---------------------------------- + --------------------------
                     2                                  2           
                                                 1 + tan (1)        
$$\frac{\left(- \sin^{2}{\left(\frac{a}{2} \right)} + \cos^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} - \frac{\left(- \frac{1}{\frac{1}{\tan^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin{\left(a \right)}}{2} - 2$$
            1                  1                                          
     ---------------- - ----------------                                  
              2                2/    pi\                                  
           sec (1)          sec |1 - --|   /   1           1      \       
     1 + ------------           \    2 /   |------- - ------------|*sec(1)
            2/    pi\   1 + ------------   |   2/a\      2/a   pi\|       
         sec |1 - --|            2         |sec |-|   sec |- - --||       
             \    2 /         sec (1)      \    \2/       \2   2 //       
-2 - ----------------------------------- + -------------------------------
                     /    pi\               /         2      \            
                2*sec|a - --|               |      sec (1)   |    /    pi\
                     \    2 /               |1 + ------------|*sec|1 - --|
                                            |       2/    pi\|    \    2 /
                                            |    sec |1 - --||            
                                            \        \    2 //            
$$\frac{\left(- \frac{1}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(1 \right)}}{\left(1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}\right) \sec{\left(1 - \frac{\pi}{2} \right)}} - 2 - \frac{- \frac{1}{\frac{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}{\sec^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}}}{2 \sec{\left(a - \frac{\pi}{2} \right)}}$$
         /       2/a\\                 /       2   \    /a\   
         |1 - tan |-||*tan(1)          \1 - tan (1)/*tan|-|   
         \        \2//                                  \2/   
-2 + --------------------------- - ---------------------------
     /       2   \ /       2/a\\   /       2   \ /       2/a\\
     \1 + tan (1)/*|1 + tan |-||   \1 + tan (1)/*|1 + tan |-||
                   \        \2//                 \        \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} - 2 - \frac{\left(1 - \tan^{2}{\left(1 \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
     sin(-2 + a)
-2 - -----------
          2     
$$- \frac{\sin{\left(a - 2 \right)}}{2} - 2$$
     /      1               1      \                                            
     |------------- - -------------|*sin(a)                                     
     |         4              2    |                                            
     |    4*sin (1)        sin (2) |                                            
     |1 + ---------   1 + ---------|               2    /   2/pi   a\      2/a\\
     |        2                4   |          2*sin (1)*|sin |-- + -| - sin |-||
     \     sin (2)        4*sin (1)/                    \    \2    2/       \2//
-2 - -------------------------------------- + ----------------------------------
                       2                            /         4   \             
                                                    |    4*sin (1)|             
                                                    |1 + ---------|*sin(2)      
                                                    |        2    |             
                                                    \     sin (2) /             
$$\frac{2 \left(- \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}\right) \sin^{2}{\left(1 \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \sin{\left(2 \right)}} - \frac{\left(- \frac{1}{\frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1} + \frac{1}{1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}}\right) \sin{\left(a \right)}}{2} - 2$$
        /        2/a\\                /        2   \    /a\   
        |-1 + cot |-||*cot(1)         \-1 + cot (1)/*cot|-|   
        \         \2//                                  \2/   
-2 + --------------------------- - ---------------------------
     /       2   \ /       2/a\\   /       2   \ /       2/a\\
     \1 + cot (1)/*|1 + cot |-||   \1 + cot (1)/*|1 + cot |-||
                   \        \2//                 \        \2//
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(1 \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} - 2 - \frac{\left(-1 + \cot^{2}{\left(1 \right)}\right) \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
     /       1                  1        \    /    pi\                                       
     |---------------- - ----------------|*cos|a - --|                                       
     |       2/    pi\            2      |    \    2 /                                       
     |    cos |1 - --|         cos (1)   |                                                   
     |        \    2 /   1 + ------------|                                                   
     |1 + ------------          2/    pi\|               /   2/a\      2/a   pi\\    /    pi\
     |         2             cos |1 - --||               |cos |-| - cos |- - --||*cos|1 - --|
     \      cos (1)              \    2 //               \    \2/       \2   2 //    \    2 /
-2 - ------------------------------------------------- + ------------------------------------
                             2                                /       2/    pi\\             
                                                              |    cos |1 - --||             
                                                              |        \    2 /|             
                                                              |1 + ------------|*cos(1)      
                                                              |         2      |             
                                                              \      cos (1)   /             
$$\frac{\left(\cos^{2}{\left(\frac{a}{2} \right)} - \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}\right) \cos{\left(1 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}\right) \cos{\left(1 \right)}} - \frac{\left(- \frac{1}{\frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}} + 1} + \frac{1}{1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}}\right) \cos{\left(a - \frac{\pi}{2} \right)}}{2} - 2$$
               /    pi\             /    pi\
     cos(a)*cos|2 - --|   cos(2)*cos|a - --|
               \    2 /             \    2 /
-2 + ------------------ - ------------------
             2                    2         
$$\frac{\cos{\left(a \right)} \cos{\left(2 - \frac{\pi}{2} \right)}}{2} - \frac{\cos{\left(2 \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{2} - 2$$
             1                   1                                                
     ----------------- - -----------------                                        
            2/     pi\             2                                              
         csc |-1 + --|          csc (1)      /     1            1   \    /     pi\
             \     2 /   1 + -------------   |------------ - -------|*csc|-1 + --|
     1 + -------------          2/     pi\   |   2/pi   a\      2/a\|    \     2 /
               2             csc |-1 + --|   |csc |-- - -|   csc |-||             
            csc (1)              \     2 /   \    \2    2/       \2//             
-2 - ------------------------------------- + -------------------------------------
                    2*csc(a)                       /       2/     pi\\            
                                                   |    csc |-1 + --||            
                                                   |        \     2 /|            
                                                   |1 + -------------|*csc(1)     
                                                   |          2      |            
                                                   \       csc (1)   /            
$$\frac{\left(\frac{1}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(-1 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}\right) \csc{\left(1 \right)}} - 2 - \frac{- \frac{1}{\frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1} + \frac{1}{1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}}}{2 \csc{\left(a \right)}}$$
              1                      1          
-2 + -------------------- - --------------------
                 /    pi\               /    pi\
     2*sec(a)*sec|2 - --|   2*sec(2)*sec|a - --|
                 \    2 /               \    2 /
$$-2 - \frac{1}{2 \sec{\left(2 \right)} \sec{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(a \right)} \sec{\left(2 - \frac{\pi}{2} \right)}}$$
              1                       1          
-2 + -------------------- - ---------------------
                 /pi    \               /     pi\
     2*csc(2)*csc|-- - a|   2*csc(a)*csc|-2 + --|
                 \2     /               \     2 /
$$-2 + \frac{1}{2 \csc{\left(2 \right)} \csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(a \right)} \csc{\left(-2 + \frac{\pi}{2} \right)}}$$
                                 /     1           2   \       
                                 |----------- - sin (1)|*sin(a)
                                 |       2             |       
     -sin(-2 + a) + sin(2 + a)   \1 + tan (1)          /       
-2 + ------------------------- - ------------------------------
                 4                             2               
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \frac{\left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin{\left(a \right)}}{2} - 2$$
     -sin(-2 + a) + sin(2 + a)      2/a\ /     1           2   \    /a\
-2 + ------------------------- - cos |-|*|----------- - sin (1)|*tan|-|
                 4                   \2/ |       2             |    \2/
                                         \1 + tan (1)          /       
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)} - 2$$
           1      
-2 - -------------
     2*csc(-2 + a)
$$-2 - \frac{1}{2 \csc{\left(a - 2 \right)}}$$
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
           2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
               /    pi\             /    pi\
     sin(2)*sin|a + --|   sin(a)*sin|2 + --|
               \    2 /             \    2 /
-2 + ------------------ - ------------------
             2                    2         
$$- \frac{\sin{\left(a \right)} \sin{\left(\frac{\pi}{2} + 2 \right)}}{2} + \frac{\sin{\left(2 \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{2} - 2$$
-2 + sin(2)*sin(a + pi/2)/2 - sin(a)*sin(2 + pi/2)/2
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