Question

Solve trigonometric function

cos2∅ + sin∅ × csc∅ / sin2∅

Answer 1

`\cot(\theta)`

Explanation:

Trig identities:

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

`sin^2(\theta)+cos^2(\theta)=1`

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

`\implies \cos(2\theta)=2cos^2(\theta)-1`

`\implies2cos^2(\theta)= \cos(2\theta)+1`

`\sin(2\theta)=2\sin(\theta)\cos(\theta)`

Therefore,

`(\cos(2\theta)+\sin(\theta) * \csc(\theta))/(\sin(2\theta))`

`=(\cos(2\theta)+(\sin(\theta))/(\sin(\theta)))/(\sin(2\theta))`

`=(\cos(2\theta)+1)/(\sin(2\theta))`

`=(2cos^2(\theta))/(\sin(2\theta))`

`=(2cos^2(\theta))/(2\sin(\theta)\cos(\theta))`

`=(cos(\theta))/(\sin(\theta))`

`=\cot(\theta)`