Question

How do you simplify this expression step by step?

Answer 1

Answer:

cot Ф

Step-by-step explanation:

Recall that sin²Ф + cos²Ф = 1, (which also says that cos²Ф - 1 = sin²Ф).

Also recall the definitions of the csc, sin and cos functions.

Your expression is equivalent to:

   1               sin Ф

----------  -  -------------

sin Ф              1

===================

           cos Ф

There are three terms in your expression:  csc, sin and cos.  Multiply all of them by sin Ф.  The result should be:

 1 - sin²Ф

---------------

sin Ф · cos Ф

Using the Pythagorean identity (see above), this simplifies to

  cos²Ф

------------------

sin Ф·cos Ф

and this whole fraction reduces to

   cos Ф

--------------   and this ratio is the definition of the cot function.

   sin Ф

Thus, the original expression is equivalent to cot Ф

Answer 2

$\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{csc(\theta )-sin(\theta )}{cos(\theta )}\implies \cfrac{~~\frac{1}{sin(\theta )}-sin(\theta )~~}{cos(\theta )}\implies \cfrac{~~\frac{1-sin^2(\theta )}{sin(\theta )}~~}{cos(\theta )}$

$\bf \cfrac{1-sin^2(\theta )}{sin(\theta )}\cdot \cfrac{1}{cos(\theta )}\implies \cfrac{\stackrel{cos(\theta )}{\begin{matrix} cos^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}} }{sin(\theta )}\cdot \cfrac{1}{\begin{matrix} cos(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} }\implies \cfrac{cos(\theta )}{sin(\theta )}\implies cot(\theta )$