Prove the identity \(\frac{\tan x + 1}{\sin x \tan x + \cos x} \equiv \sin x + \cos x\).
Solution
Let \(s = \sin x\) and \(c = \cos x\). Then \(\tan x = \frac{s}{c}\).
Substitute \(\tan x = \frac{s}{c}\) into the left-hand side:
\(\frac{\frac{s}{c} + 1}{s \cdot \frac{s}{c} + c} = \frac{\frac{s+c}{c}}{\frac{s^2}{c} + c}\)
Simplify the expression:
\(= \frac{s+c}{s^2 + c^2}\)
Since \(s^2 + c^2 = 1\), the expression becomes:
\(= s + c\)
This matches the right-hand side, \(\sin x + \cos x\).
Log in to record attempts.