Question

In the first step of a proof, the left-hand side of the following identity is factored.

sine squared theta minus cosine squared theta sine squared theta equals sine superscript 4 baseline theta

Which fundamental identity is used in the second step of the proof?
(1 point)
Responses

sine theta equals start fraction 1 over cosecant theta end fraction
Image with alt text: sine theta equals start fraction 1 over cosecant theta end fraction

tangent theta equals start fraction sine theta over cosine theta end fraction
Image with alt text: tangent theta equals start fraction sine theta over cosine theta end fraction

sine squared theta plus cosine squared theta equals 1
Image with alt text: sine squared theta plus cosine squared theta equals 1

1 plus tangent squared theta equals secant squared theta

Answer 1

To determine which fundamental identity is used in the proof, let's first analyze the left-hand side of the given equation:

\[ \sin^2 \theta - \cos^2 \theta \sin^2 \theta \]

This expression can be factored as follows:

\[ \sin^2 \theta (1 - \cos^2 \theta) \]

Next, we recall the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

From this identity, we can deduce that:

\[ 1 - \cos^2 \theta = \sin^2 \theta \]

Substituting this back into the factored expression gives:

\[ \sin^2 \theta \cdot \sin^2 \theta = \sin^4 \theta \]

Thus, the identity used in the second step of the proof is:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

So the correct answer is:

**sine squared theta plus cosine squared theta equals 1**.