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Solve trigonometric function

sin∅ × csc∅ + cos(3π/2 - ∅) × sin∅

Sagot :

fieryanswererft

Answer:

[tex]\sf \cos ^2\left(x\right)[/tex]

Explanation:

[tex]\sf \sin \left(x\right)\csc \left(x\right)+\cos \left(\frac{3\pi }{2}-x\right)\sin \left(x\right)[/tex]

[tex]\sf \sin \left(x\right)\csc \left(x\right)+\left(-\sin \left(x\right)\right)\sin \left(x\right)[/tex]

[tex]\sf \sin \left(x\right)\csc \left(x\right)-\sin ^2\left(x\right)[/tex]

[tex]\sf \cos ^2\left(x\right)[/tex]

semsee45

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]

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