Question

How do you simplify this expression step by step?

Answer 1

cot Ф

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

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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{~~(1)/(sin(\theta ))-sin(\theta )~~}{cos(\theta )}\implies \cfrac{~~(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 )`