Question

Simplify the expression using trigonometric identities (csc θ – csc θ · cos2 θ).A) sin2 θB) sin θ · tan θC) sin3 θD) sin θ

Answer 1

Given the expression;

`\csc \theta-\csc \theta\cdot\cos ^2\theta`

This can be simplified as;

`\csc \theta-\csc \theta\cdot\cos ^2\theta=\csc \theta(1-\cos ^2\theta)`

Recall the identity that;

`\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \sin ^2\theta=1-\cos ^2\theta \end{gathered}`

Then, we have;

`\csc \theta-\csc \theta\cdot\cos ^2\theta=\csc \theta(\sin ^2\theta)`

Also, recall that;

`\csc \theta=(1)/(\sin \theta)`

So, we have;

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

CORRECT OPTION: D