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Solve trigonometric function

cos2∅ + sin∅ × csc∅ / sin2∅

Sagot :

semsee45

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]

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