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Answer:
[tex]\cot(\theta)[/tex]
Step-by-step explanation:
Trig identities:
[tex]\csc(\theta)=\dfrac{1}{\sin(\theta)}[/tex]
[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]
[tex]\cos(2\theta)=cos^2(\theta)-sin^2(\theta)[/tex]
[tex]\implies \cos(2\theta)=2cos^2(\theta)-1[/tex]
[tex]\implies2cos^2(\theta)= \cos(2\theta)+1[/tex]
[tex]\sin(2\theta)=2\sin(\theta)\cos(\theta)[/tex]
Therefore,
[tex]\dfrac{\cos(2\theta)+\sin(\theta) \times \csc(\theta)}{\sin(2\theta)}[/tex]
[tex]=\dfrac{\cos(2\theta)+\dfrac{\sin(\theta)}{\sin(\theta)}}{\sin(2\theta)}[/tex]
[tex]=\dfrac{\cos(2\theta)+1}{\sin(2\theta)}[/tex]
[tex]=\dfrac{2cos^2(\theta)}{\sin(2\theta)}[/tex]
[tex]=\dfrac{2cos^2(\theta)}{2\sin(\theta)\cos(\theta)}[/tex]
[tex]=\dfrac{cos(\theta)}{\sin(\theta)}[/tex]
[tex]=\cot(\theta)[/tex]