Question

Solve trigonometric function sin∅ × csc∅ + cos(3π/2 - ∅) × sin∅

Answer 1

Answer:

$\sf \cos ^2\left(x\right)$

Explanation:

$\sf \sin \left(x\right)\csc \left(x\right)+\cos \left(\frac{3\pi }{2}-x\right)\sin \left(x\right)$

$\sf \sin \left(x\right)\csc \left(x\right)+\left(-\sin \left(x\right)\right)\sin \left(x\right)$

$\sf \sin \left(x\right)\csc \left(x\right)-\sin ^2\left(x\right)$

$\sf \cos ^2\left(x\right)$

Answer 2

Answer:

$\cos^2(\theta)$

Step-by-step explanation:

Identities used:

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

$\cos(\frac{3 \pi}{2}-\theta)=\cos(\frac{3 \pi}{2})\cos(\theta)+\sin(\frac{3 \pi}{2})\sin(\theta)$

$\textsf{As }\cos(\frac{3 \pi}{2})=0\textsf{ and }\sin(\frac{3 \pi}{2})=-1$

$\implies \cos(\frac{3 \pi}{2}-\theta)=0 \times\cos(\theta)+-1\times\sin(\theta)=-\sin(\theta)$

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

Therefore,

$\sin(\theta) \times \csc(\theta)+\cos(\frac{3 \pi}{2}-\theta)\times\sin(\theta)$

$=\sin(\theta) \times\dfrac{1}{\sin(\theta)}-\sin(\theta)\times\sin(\theta)$

$=1-\sin^2(\theta)$

$= \cos^2(\theta)$