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Chapter 6: Problem 83

Show that $$ \sin ^{2}(2 \theta)=4\left(\sin ^{2} \theta-\sin ^{4} \theta\right) $$ for all \(\theta\).

Short Answer

Expert verified
Using the double angle formula for sine, we substitute the expression for \(\sin(2\theta)\) in the given identity and expand it to get \(\sin^2(2\theta) = 4\sin^2(\theta)\cos^2(\theta)\). We then use the Pythagorean trigonometric identity to substitute for \(\cos^2(\theta)\) and simplify the expression, ultimately showing that \(\sin^2(2\theta) = 4\left(\sin^2(\theta) - \sin^4(\theta)\right)\) for all \(\theta\).

Step by step solution

01

Use the double angle formula

Start by using the double angle formula for sine to substitute the expression for \(\sin(2\theta)\) in the given identity: \(\sin ^{2}(2 \theta)=\left(2\sin(\theta)\cos(\theta)\right)^2\)
02

Expand the expression

Expand the expression in Step 1: \(\sin ^{2}(2 \theta)=4\sin^2(\theta)\cos^2(\theta)\)
03

Use the Pythagorean trigonometric identity

Since we know that \(\sin^2(\theta) + \cos^2(\theta) = 1\), we can solve for \(\cos^2(\theta)\) and substitute into our expression: \(\cos^2(\theta) = 1 - \sin^2(\theta)\) Now, substitute this expression into our expanded expression from Step 2: \(\sin ^{2}(2 \theta)=4\sin^2(\theta)(1 - \sin^2(\theta))\)
04

Simplify the expression

Finally, simplify the expression from Step 3 to obtain the desired identity: \(\sin ^{2}(2 \theta)=4\left(\sin^2(\theta) - \sin^4(\theta)\right)\) We have now shown that the given trigonometric identity holds for all \(\theta\).

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