Question

The expression cot theta - csc^2theta/cot theta simplifies to what expression?

Answer 1

-tan θ

EXPLANATION :

Note that :

`\begin{gathered} \cot\theta=(\cos\theta)/(\sin\theta)\quad,\quad\csc\theta=(1)/(\sin\theta) \\ \\ -\sin^2=\cos^2\theta-1 \end{gathered}`

From the problem, we have :

`\cot\theta-(\csc^2\theta)/(\cot\theta)`

Subsitute the identities :

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

The answer is -tan θ