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Question

# Verify that each trigonometric equation is an identity. $\sin ^{2} \beta\left(1+\cot ^{2} \beta\right)=1$

Solution

Verified
Step 1
1 of 2

Work on the more complex side, which is the left side.

Use the Pythagorean identity: $1+\cot^2\beta=\csc^2\beta$

$\sin^{2}\beta\left(1+\cot^{2}\beta\right)=\sin^{2}\beta\left(\csc^{2}\beta\right)$

Use the reciprocal identity: $\csc\beta=\dfrac{1}{\sin\beta}$

$\sin^{2}\beta\left(1+\cot^{2}\beta\right)=\sin^{2}\beta\cdot\dfrac{1}{\csc^{2}\beta}$

$\color{#c34632}\sin^{2}\beta\left(1+\cot^{2}\beta\right)=1\hspace{2mm}\checkmark\color{white}\tag{1}$

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