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Chapter 6: Problem 47

Show that $$ \sin u \sin v=\frac{\cos (u-v)-\cos (u+v)}{2} $$ for all \(u, v\).

Short Answer

Expert verified
Using the cosine addition and subtraction formulas, we obtain \(\cos (u + v) = \cos u \cos v - \sin u \sin v\) and \(\cos (u - v) = \cos u \cos v + \sin u \sin v\). Subtracting the second equation from the first, we get \(\cos (u + v) - \cos (u - v) = - 2 \sin u \sin v\). Dividing both sides by -2, we arrive at the desired trigonometric identity: \(\sin u \sin v = \frac{\cos (u - v) - \cos (u + v)}{2}\).

Step by step solution

01

Utilize addition formula for cosine

Recall the cosine addition formula: \[ \cos (A + B) = \cos A \cos B - \sin A \sin B \] Similarly, the cosine subtraction formula is: \[ \cos (A - B) = \cos A \cos B + \sin A \sin B \]
02

Replace A and B with u and v

Replace A with u and B with v in both the addition and subtraction formulas, obtaining: \[ \cos (u + v) = \cos u \cos v - \sin u \sin v \] \[ \cos (u - v) = \cos u \cos v + \sin u \sin v \]
03

Subtract the two equations

Subtract the second equation from the first: \[ \cos (u + v) - \cos (u - v) = (\cos u \cos v - \sin u \sin v) - (\cos u \cos v + \sin u \sin v) \]
04

Simplify the equation

Simplify the right-hand side of the equation by combining like terms: \[ \cos (u + v) - \cos (u - v) = - 2 \sin u \sin v \]
05

Solve for the desired expression

Divide both sides by -2 to obtain the identity we were asked to prove: \[ \sin u \sin v = \frac{\cos (u - v) - \cos (u + v)}{2} \] Thus, we have shown that the given identity holds true for all values of u and v.

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