Question

Solve trigonometric function

sin∅ × csc∅ + cos(3π/2 - ∅) × sin∅

Answer 1

`\cos^2(\theta)`

Step-by-step explanation:

Identities used:

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

`\cos((3 \pi)/(2)-\theta)=\cos((3 \pi)/(2))\cos(\theta)+\sin((3 \pi)/(2))\sin(\theta)`

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

`\implies \cos((3 \pi)/(2)-\theta)=0 *\cos(\theta)+-1*\sin(\theta)=-\sin(\theta)`

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

Therefore,

`\sin(\theta) * \csc(\theta)+\cos((3 \pi)/(2)-\theta)*\sin(\theta)`

`=\sin(\theta) *(1)/(\sin(\theta))-\sin(\theta)*\sin(\theta)`

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

`= \cos^2(\theta)`

Answer 2

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

Step-by-step explanation:

`\sf \sin \left(x\right)\csc \left(x\right)+\cos \left((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)`