Question

Please Help!!! 50 Points!!!
In the first step of a proof, the left-hand side of the following identity is factored.
sin^2∅-cos^2∅sin^2∅=sin^4∅

Answer 1

The fundamental identity used in the second step of the proof is sin^2(θ) + cos^2(θ) = 1

Answer 2

An identity is true for general case, and not only for special cases. The proof for given statement is derived by using Pythagorean identity.

What are Pythagorean identities ?

$sin^2(\theta) + cos^2(\theta) = 1\\\\1 + tan^2(\theta) = sec^2(\theta)\\\\1 + cot^2(\theta) = csc^2(\theta)$

Given statement is $\sin^2\theta - \cos^2\theta\sin^2\theta =\sin^4\theta$

Taking its left side:

$\begin{aligned}\sin^2\theta(1 - cos^2\theta) &= \sin^2\theta \times sin^2\theta \\&= sin^4\theta \end{aligned}$

(from first Pythagorean identity).

Thus, the given statement is proved using Pythagorean identity (first) that $sin^2\theta + cos^2\theta = 1$

Learn more about first Pythagorean Identity here:

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