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N76 [4]
3 years ago
13

A. Show

Mathematics
1 answer:
nydimaria [60]3 years ago
8 0

a. Recall the double angle identities:

\sin^2x=\dfrac{1-\cos2x}2

\cos^2x=\dfrac{1+\cos2x}2

Then

\sin^2x\cos^2x=\dfrac{(1-\cos2x)(1+\cos2x)}4=\dfrac{1-\cos^22x}4

Applying the identity again, we have

\sin^2x\cos^2x=\dfrac{1-\frac{1+\cos4x}2}4=\dfrac{2-(1+\cos4x)}8=\dfrac{1-\cos4x}8

as required.

b. Using the result from part (a),

\sin^2x\cos^2x=\dfrac{1-\cos4x}8=\dfrac{2-\sqrt2}{16}

\implies\cos4x=\dfrac1{\sqrt2}

\implies4x=\dfrac\pi4+2n\pi\text{ or }4x=-\dfrac\pi4+2n\pi

(where n is any integer)

\implies\boxed{x=\pm\dfrac\pi{16}+\dfrac{n\pi}2}

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