Question

The expression tangent theta minus the quantity secant squared theta over tangent theta end quantity simplifies to what expression?−tan θ −cot θ cos θ sec θ

Answer 1

Given the expression:

To simplify, recall from trigonometric identities:

`\begin{gathered} \tan\theta=(\sin\theta)/(\cos\theta) \\ \\ sec\theta=(1)/(\cos\theta) \\ \\ \sin^2\theta+\cos^2\theta=1 \\ \Rightarrow-\cos^2\theta=\sin^2\theta-1 \\ \\ (1)/(\tan\theta)=cot\theta \end{gathered}`

This implies that

`\begin{gathered} (\sin\theta)/(\cos\theta)-((1)/(\cos^2\theta)*(\cos\theta)/(\sin\theta)) \\ =(\sin\theta)/(\cos\theta)-(1)/(\sin\theta\cos\theta) \\ =(\sin^2\theta-1)/(\sin\theta\cos\theta) \\ but\text{ -}\cos^2\theta=\sin^2\theta-1 \\ thus,\text{ we have} \\ (-\cos^2\theta)/(\sin\theta\cos\theta) \\ =-(\cos\theta)/(sin\theta) \\ =-cot\theta \end{gathered}`

Hence, the expression is simplified to be

`-cot\theta`

Hence, the correct option is