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

Answer 1

Let's analyze Sami's proof step by step to identify the mistake.

1. The expression starts with \( \cos^2 \theta \tan^2 \theta \).
2. It then equates this to \( \cos^2 \theta (\sec^2 \theta - 1) \). This part is correct since \( \sec^2 \theta - 1 = \tan^2 \theta \).
3. The next step should say that this equals \( \cos^2 \theta \tan^2 \theta \), which is true.
4. Then it tries to express \( \cos^2 \theta (\sec^2 \theta - 1) \) as \( \cos^2 \theta \sec^2 \theta - \cos^2 \theta \). This is also correct given the identity.
5. Sammy then seems to evaluate \( \cos^2 \theta \cot^2 \theta \), but here it looks like he has a mistake when transitioning to \( \cot^2 \theta \).

Now, evaluating the options:

• Option 1 and 2 discuss tangent theta being incorrect; tangent is indeed \( \frac{\sin \theta}{\cos \theta} \).
• Option 3 states secant is incorrectly defined as sine, it’s actually cosine, which is also valid.
• Option 4 addresses the factorization step—the assumption about \( \cos^2 \theta (\sec^2 \theta - 1) \) being equivalent to what he stated is indeed wrong.
• Option 5 compared cotangent incorrectly; this option is stating the same mistake in context but misleading by asserting it involves cotangent.

The correct mistake presented clearly is option 4: "He assumed \( \cos^2 \theta (\sec^2 \theta - 1) = \cos^2 \theta \sec^2 \theta - \cos^2 \theta \), but it equals \( \cos^2 \theta \sec^2 \theta - 1 \).", reveals how the misstep occurred in manipulating the equation.

So the answer is that he misapplied the identity incorrectly in this transitional step.