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Question

# Verify each identity. Give the domain of validity for each identity. $\sin \theta \tan \theta=\frac{1}{\cos \theta}-\cos \theta$

Solution

Verified

Given: $\sin\theta\tan\theta=\dfrac{1}{\cos\theta}-\cos\theta$

We work using the left side. Apply the tangent identity:

$\sin\theta\tan\theta=\sin\theta\cdot\frac{\sin\theta}{\cos\theta}$

$\sin\theta\tan\theta=\frac{\sin^2\theta}{\cos\theta}$

Apply the Pythagorean identity $(\cos^2\theta+\sin^2\theta=1)$:

$\sin\theta\tan\theta=\frac{1-\cos^2\theta}{\cos\theta}$

$\sin\theta\tan\theta=\frac{1}{\cos\theta}-\frac{\cos^2\theta}{\cos\theta}$

$\color{#c34632}{ \sin\theta\tan\theta=\frac{1}{\cos\theta}-\cos\theta}$

The domains of $\sin\theta$ and $\cos\theta$ is all real numbers. The domains of $\tan\theta$ and $\dfrac{1}{\cos\theta }$ exclude all zeros of $\cos\theta$ which are the odd multiples of $\dfrac{\pi}{2}$.

$\textbf{\color{#c34632}{Domain of validity:}}$ All real numbers except for odd multiples of $\dfrac{\pi}{2}$.

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