Answer:
[tex]\cos^2(\theta)[/tex]
Step-by-step explanation:
Identities used:
[tex]\csc(\theta)=\dfrac{1}{\sin(\theta)}[/tex]
[tex]\cos(\frac{3 \pi}{2}-\theta)=\cos(\frac{3 \pi}{2})\cos(\theta)+\sin(\frac{3 \pi}{2})\sin(\theta)[/tex]
[tex]\textsf{As }\cos(\frac{3 \pi}{2})=0\textsf{ and }\sin(\frac{3 \pi}{2})=-1[/tex]
[tex]\implies \cos(\frac{3 \pi}{2}-\theta)=0 \times\cos(\theta)+-1\times\sin(\theta)=-\sin(\theta)[/tex]
[tex]\sin^2(\theta)+\cos^2(\theta)=1 \implies \cos^2(\theta)=1-\sin^2(\theta)[/tex]
Therefore,
[tex]\sin(\theta) \times \csc(\theta)+\cos(\frac{3 \pi}{2}-\theta)\times\sin(\theta)[/tex]
[tex]=\sin(\theta) \times\dfrac{1}{\sin(\theta)}-\sin(\theta)\times\sin(\theta)[/tex]
[tex]=1-\sin^2(\theta)[/tex]
[tex]= \cos^2(\theta)[/tex]
