Sammy wrote the following proof. What did he do wrong?

Question

math expressioncosine squared theta tangent squared theta equals cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared secant squared minus cosine squared equals cosine squared theta start fraction 1 over sine squared theta end fraction minus cosine squared theta equals cotangent squared theta minus cosine squared theta
(1 point)
Responses

He assumed tangent theta equals start fraction sine theta over cosine theta end fraction, but tangent theta equals start fraction cosine theta over sine theta end fraction.
He assumed Image with alt text: tangent theta equals start fraction sine theta over cosine theta end fraction , but Image with alt text: tangent theta equals start fraction cosine theta over sine theta end fraction .

He assumed secant theta equals start fraction 1 over sine theta end fraction, but secant theta equals start fraction 1 over cosine theta end fraction.
He assumed Image with alt text: secant theta equals start fraction 1 over sine theta end fraction , but Image with alt text: secant theta equals start fraction 1 over cosine theta end fraction .

He assumed cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus cosine squared theta, but cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus 1.
He assumed Image with alt text: cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus cosine squared theta , but Image with alt text: cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus 1 .

He assumed start fraction cosine squared theta over sine squared theta end fraction equals cotangent squared theta, but start fraction cosine squared theta over sine squared theta end fraction equals tangent squared theta.

Answer 1

The most evident mistake in Sammy's proof lies in the assumption regarding secant. He assumed:

"secant squared theta - 1 = cosine squared theta secant squared theta - cosine squared theta."

The correct relationship is:

\[ \secant^2 \theta - 1 = \tan^2 \theta \]

In another part, it seems like Sammy is misapplying the relationships involving trigonometric identities.

However, the specific issue highlighted is that he assumed:

"He assumed cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus cosine squared theta, but cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus 1."

This means that he incorrectly expanded the term \( \cos^2 \theta (\sec^2 \theta - 1) \) to equal \( \cos^2 \theta \sec^2 \theta - \cos^2 \theta \) instead of recognizing that it represents another trigonometric relation.

Thus, the correct option addressing the mistake in Sammy's proof is:

He assumed cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus cosine squared theta, but cosine squared theta left parenthesis secant squared theta minus 1 right parenthesis equals cosine squared theta secant squared theta minus 1.