Mister Exam
Least common denominator cos(a)/2*sin(2)-1-sin(a)/2*cos(2)-1
The solution
You have entered
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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
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sin(-2 + a)
-2 - -----------
2
$$- \frac{\sin{\left(a - 2 \right)}}{2} - 2$$
-2 - sin(-2 + a)/2
Combining rational expressions
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-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
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-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
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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
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-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
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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
Trigonometric part
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/ 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