Is sin^2(x)cos^2(x)= 1/4 sin^2(2x) ?

The answer to whether sin^2(x)cos^2(x)= 1/4 sin^2(2x) is True

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prove sin^2(x)cos^2(x)= 1/4 sin^2(2x)

prove

sin^{2} (x) cos^{2} (x) = \frac{1}{4} sin^{2} (2 x)

True

Solution steps

sin^{2} (x) cos^{2} (x) = \frac{1}{4} sin^{2} (2 x)

Manipulating right side

\frac{1}{4} sin^{2} (2 x)

Rewrite using trig identities

\frac{1}{4} sin^{2} (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= \frac{1}{4} (2 sin (x) cos (x))^{2}

Simplify

\frac{1}{4} (2 sin (x) cos (x))^{2} : sin^{2} (x) cos^{2} (x)

\frac{1}{4} (2 sin (x) cos (x))^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( 2 sin ( x ) cos ( x ) ) ^{2}}{4}

Multiply:

1 \cdot (2 sin (x) cos (x))^{2} = (2 sin (x) cos (x))^{2}

= \frac{( 2 sin ( x ) cos ( x ) ) ^{2}}{4}

(2 sin (x) cos (x))^{2} = 2^{2} sin^{2} (x) cos^{2} (x)

(2 sin (x) cos (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} sin^{2} (x) cos^{2} (x)

= \frac{2 ^{2} sin ^{2} ( x ) cos ^{2} ( x )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{2} sin ^{2} ( x ) cos ^{2} ( x )}{2 ^{2}}

Cancel the common factor:

2^{2}

= sin^{2} (x) cos^{2} (x)

= sin^{2} (x) cos^{2} (x)

= sin^{2} (x) cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

Is sin^2(x)cos^2(x)= 1/4 sin^2(2x) ?
The answer to whether sin^2(x)cos^2(x)= 1/4 sin^2(2x) is True