Mister Exam
Least common denominator cos(x)/x-sin(x)/x^2
The solution
You have entered
[src]
cos(x) sin(x)
------ - ------
x 2
x
$$- \frac{\sin{\left(x \right)}}{x^{2}} + \frac{\cos{\left(x \right)}}{x}$$
cos(x)/x - sin(x)/x^2
General simplification
[src]
-sin(x) + x*cos(x)
------------------
2
x
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$$
(-sin(x) + x*cos(x))/x^2
Rational denominator
[src]
2
x *cos(x) - x*sin(x)
--------------------
3
x
$$\frac{x^{2} \cos{\left(x \right)} - x \sin{\left(x \right)}}{x^{3}}$$
(x^2*cos(x) - x*sin(x))/x^3
Common denominator
[src]
-sin(x) + x*cos(x)
------------------
2
x
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$$
(-sin(x) + x*cos(x))/x^2
Combining rational expressions
[src]
-sin(x) + x*cos(x)
------------------
2
x
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$$
(-sin(x) + x*cos(x))/x^2
Combinatorics
[src]
-sin(x) + x*cos(x)
------------------
2
x
$$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$$
(-sin(x) + x*cos(x))/x^2
Powers
[src]
I*x -I*x
e e
---- + ----- / -I*x I*x\
2 2 I*\- e + e /
------------ + ------------------
x 2
2*x
$$\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{x} + \frac{i \left(e^{i x} - e^{- i x}\right)}{2 x^{2}}$$
cos(x) sin(x)
------ - ------
x 2
x
$$\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}$$
cos(x)/x - sin(x)/x^2
Assemble expression
[src]
cos(x) sin(x)
------ - ------
x 2
x
$$\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}$$
cos(x)/x - sin(x)/x^2
Trigonometric part
[src]
2/x\ /x\
1 - tan |-| 2*tan|-|
\2/ \2/
----------- - ---------------
2/x\ / 2/x\\
1 + tan |-| x*|1 + tan |-||
\2/ \ \2//
-----------------------------
x
$$\frac{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{x}$$
1 1
----------- - --------
/pi \ x*csc(x)
csc|-- - x|
\2 /
----------------------
x
$$\frac{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{x \csc{\left(x \right)}}}{x}$$
2/x\ /x\
-1 + cot |-| 2*cot|-|
\2/ \2/
------------ - ---------------
2/x\ / 2/x\\
1 + cot |-| x*|1 + cot |-||
\2/ \ \2//
------------------------------
x
$$\frac{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{x}$$
sin(x)
- ------ + cos(x)
x
-----------------
x
$$\frac{\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}}{x}$$
/ pi\
sin|x + --|
\ 2 / sin(x)
----------- - ------
x 2
x
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}$$
1 1
------------- - ---------
/pi \ 2
x*csc|-- - x| x *csc(x)
\2 /
$$\frac{1}{x \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{x^{2} \csc{\left(x \right)}}$$
2/x\ /x\
1 - tan |-| 2*tan|-|
\2/ \2/
--------------- - ----------------
/ 2/x\\ 2 / 2/x\\
x*|1 + tan |-|| x *|1 + tan |-||
\ \2// \ \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{x^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ pi\
cos|x - --|
\ 2 /
- ----------- + cos(x)
x
----------------------
x
$$\frac{\cos{\left(x \right)} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x}}{x}$$
1 1
-------- - --------------
x*sec(x) 2 / pi\
x *sec|x - --|
\ 2 /
$$\frac{1}{x \sec{\left(x \right)}} - \frac{1}{x^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
sin(x) / pi\
- ------ + sin|x + --|
x \ 2 /
----------------------
x
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)} - \frac{\sin{\left(x \right)}}{x}}{x}$$
cos(x) sin(x)
------ - ------
x 2
x
$$\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}$$
2/x\ /x\
-1 + cot |-| 2*cot|-|
\2/ \2/
--------------- - ----------------
/ 2/x\\ 2 / 2/x\\
x*|1 + cot |-|| x *|1 + cot |-||
\ \2// \ \2//
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{x^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
1 1
-------- - ---------
x*sec(x) 2
x *csc(x)
$$\frac{1}{x \sec{\left(x \right)}} - \frac{1}{x^{2} \csc{\left(x \right)}}$$
1 1
------ - -------------
sec(x) / pi\
x*sec|x - --|
\ 2 /
----------------------
x
$$\frac{\frac{1}{\sec{\left(x \right)}} - \frac{1}{x \sec{\left(x - \frac{\pi}{2} \right)}}}{x}$$
/ pi\
cos|x - --|
cos(x) \ 2 /
------ - -----------
x 2
x
$$\frac{\cos{\left(x \right)}}{x} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x^{2}}$$
cos(x)/x - cos(x - pi/2)/x^2