Question

Solve trigonometric function cos2∅ + sin∅ × csc∅ / sin2∅

Answer 1

Answer:

$\cot(\theta)$

Step-by-step explanation:

Trig identities:

$\csc(\theta)=\dfrac{1}{\sin(\theta)}$

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

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

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

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

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

Therefore,

$\dfrac{\cos(2\theta)+\sin(\theta) \times \csc(\theta)}{\sin(2\theta)}$

$=\dfrac{\cos(2\theta)+\dfrac{\sin(\theta)}{\sin(\theta)}}{\sin(2\theta)}$

$=\dfrac{\cos(2\theta)+1}{\sin(2\theta)}$

$=\dfrac{2cos^2(\theta)}{\sin(2\theta)}$

$=\dfrac{2cos^2(\theta)}{2\sin(\theta)\cos(\theta)}$

$=\dfrac{cos(\theta)}{\sin(\theta)}$

$=\cot(\theta)$

Answer 2

Answer:

Step-by-step explanation:

$\sf Formula: \\\\\sin 2 \theta = 2 \sin \theta \cos \theta$


$\text {\sf \Large Solution :}$

$\cos 2 \theta +\dfrac{\sin \theta\cdot \cos \theta }{\sin 2 \theta } = \\\\\ \cos 2 \theta +\dfrac{\sin \theta\cdot \cos \theta}{2\sin \theta\cdot \cos \theta} =\boxed{\cos 2 \theta +\dfrac{1}{2} }$