Question 1

Solve trigonometric function

cos2∅ + sin∅ × csc∅ / sin2∅

Question 2

Answer:

$\cot(\theta)$

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,

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

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

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

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

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

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

$=\cot(\theta)$

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