Is sin^4(θ)+cos^2(θ)sin^2(θ)=sin^2(θ) ?

The answer to whether sin^4(θ)+cos^2(θ)sin^2(θ)=sin^2(θ) is True

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prove sin^4(θ)+cos^2(θ)sin^2(θ)=sin^2(θ)

prove

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ) = sin^{2} (θ)

True

Solution steps

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ) = sin^{2} (θ)

Manipulating left side

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ)

Factor

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ) : sin^{2} (θ) (sin^{2} (θ) + cos^{2} (θ))

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ)

Apply exponent rule:

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

sin^{4} (θ) = sin^{2} (θ) sin^{2} (θ)

= sin^{2} (θ) sin^{2} (θ) + cos^{2} (θ) sin^{2} (θ)

Factor out common term

sin^{2} (θ)

= sin^{2} (θ) (sin^{2} (θ) + cos^{2} (θ))

= (cos^{2} (θ) + sin^{2} (θ)) sin^{2} (θ)

Rewrite using trig identities

(cos^{2} (θ) + sin^{2} (θ)) sin^{2} (θ)

Use the Pythagorean identity:

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

= 1 \cdot sin^{2} (θ)

Simplify

= sin^{2} (θ)

= sin^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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