Question

This one is a proof please i need help

Answer 1

See below for proof using trigonometric identities.

Explanation:

`\boxed{\begin{minipage}{5 cm}\underline{Trigonometric Identities}\\\\$\tan \theta=(\sin \theta)/(\cos \theta)$\\\\\\$\cot \theta=(\cos \theta)/(\sin \theta)$\\\\\\$\csc \theta=(1)/(\sin \theta)$\\\\\\$\sin^2 \theta+\cos^2 \theta=1$\\\end{minipage}}`

To prove the given equation, use trigonometric identities to rewrite the left side of the equation.

`\begin{aligned} (\cot x)/(\csc x)-(\csc x)/(\cot x)&=((\cos x)/(\sin x))/((1)/(\sin x))-((1)/(\sin x))/((\cos x)/(\sin x))\\\\&=(\cos x)/(1)-(1)/(\cos x)\\\\&=(\cos x)/(1) \cdot (\cos x)/(\cos x)-(1)/(\cos x)\\\\&=(\cos^2 x-1)/(\cos x)\\\\&=(\cos^2 x-(\sin^2 x + \cos^2 x))/(\cos x)\\\\&=(-\sin^2 x)/(\cos x)\\\\&=-(\sin x )/(\cos x) \cdot (\sin x)/(1)\\\\&=-\tan x \sin x\end{aligned}`

Hence proving that:

`(\cot x)/(\csc x)-(\csc x)/(\cot x)=-\tan x \sin x`