Problem 67 - Precalculus Essentials

Question

Rewrite each expression in terms of the given function or functions. $$\frac{\tan x+\cot x}{\csc x} ; \cos x$$

Step-by-Step Solution

Answer

The expression \(\frac{\tan x + \cot x}{\csc x}\) rewritten in terms of \(\cos x\) is \(\frac{1}{\sqrt{1 - \cos^2 x}}\).

Step 1: Rewrite in terms of sin and cos

Firstly, rewrite \(\tan x\), \(\cot x\), and \(\csc x\) in terms of \(\sin x\) and \(\cos x\). Therefore, the expression \(\frac{\tan x+\cot x}{\csc x}\) becomes \(\frac{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}}{\frac{1}{\sin x}}\).

Step 2: Simplify the expression

Simplify the expression in the numerator. After simplifying, the resulting expression is \(\frac{\sin^2 x + \cos^2 x}{\sin x}\). The expression \(\sin^2 x + \cos^2 x\) is an identity that equals 1, so rewrite it as such.

Step 3: Substitute the identity

Replace the \(\sin^2 x + \cos^2 x\) with 1. The expression therefore becomes \(\frac{1}{\sin x}\). This, by definition, equals \(\csc x\).

Step 4: Rewrite in terms of cos

Lastly, remember that \(\csc x = \frac{1}{\sin x} = \frac{1}{\sqrt{1 - \cos^2 x}}\). This is because \(\sin^2 x + \cos^2 x = 1\), so \(\sin x = \sqrt{1 - \cos^2 x}\). Therefore our final result is \(\frac{1}{\sqrt{1 - \cos^2 x}}\).