verify the identity.\n\\( \\frac { \\sin ( \\alpha + \\beta ) } { \\sin \\alpha \\sin \\beta } = \\cot…

verify the identity.\n\\( \\frac { \\sin ( \\alpha + \\beta ) } { \\sin \\alpha \\sin \\beta } = \\cot \\beta + \\cot \\alpha \\)\nwrite the left side numerator in terms of a sum or difference formula for sine or cosine.\n\\( \\frac { \\sin ( \\alpha + \\beta ) } { \\sin \\alpha \\sin \\beta } = \\frac { } { \\sin \\alpha \\sin \\beta } \\)

verify the identity.\n\\( \\frac { \\sin ( \\alpha + \\beta ) } { \\sin \\alpha \\sin \\beta } = \\cot \\beta + \\cot \\alpha \\)\nwrite the left side numerator in terms of a sum or difference formula for sine or cosine.\n\\( \\frac { \\sin ( \\alpha + \\beta ) } { \\sin \\alpha \\sin \\beta } = \\frac { } { \\sin \\alpha \\sin \\beta } \\)

Answer

Explanation:

Step1: Use the sum formula for sine

The sum formula for sine is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. So, $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$.

Step2: Divide by $\sin\alpha\sin\beta$

Divide each term in the numerator $\sin\alpha\cos\beta+\cos\alpha\sin\beta$ by $\sin\alpha\sin\beta$. We get $\frac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\sin\alpha\sin\beta}=\frac{\sin\alpha\cos\beta}{\sin\alpha\sin\beta}+\frac{\cos\alpha\sin\beta}{\sin\alpha\sin\beta}$.

Step3: Simplify each fraction

Using the identity $\cot x=\frac{\cos x}{\sin x}$, $\frac{\sin\alpha\cos\beta}{\sin\alpha\sin\beta}=\cot\beta$ and $\frac{\cos\alpha\sin\beta}{\sin\alpha\sin\beta}=\cot\alpha$.

Answer:

$\cot\beta+\cot\alpha$