Is sin^2(x)cos^2(x)=sin^2(x)-sin^4(x) ?

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

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prove sin^2(x)cos^2(x)=sin^2(x)-sin^4(x)

prove

sin^{2} (x) cos^{2} (x) = sin^{2} (x) - sin^{4} (x)

True

Solution steps

sin^{2} (x) cos^{2} (x) = sin^{2} (x) - sin^{4} (x)

Manipulating right side

sin^{2} (x) - sin^{4} (x)

Factor

sin^{2} (x) - sin^{4} (x) : sin^{2} (x) (1 - sin^{2} (x))

sin^{2} (x) - sin^{4} (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{4} (x) = sin^{2} (x) sin^{2} (x)

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

Factor out common term

sin^{2} (x)

= sin^{2} (x) (1 - sin^{2} (x))

= (1 - sin^{2} (x)) sin^{2} (x)

Rewrite using trig identities

(1 - sin^{2} (x)) sin^{2} (x)

Use the Pythagorean identity:

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin (x) + cos (x) = \frac{1}{cot ( x )}

\frac{sin ( x )}{1 - cos ( x )} = 1 + cos (x)

1 + cos (a) = 2 cos^{2} (\frac{a}{2})

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