Julia wants to simplify the term start fraction secant squared theta minus 1 over cotangent squared theta plus 1 end fraction in a trigonometric identity that she is proving. Which of the following identities should she use to help her? Select all that apply.

Question

(1 point)
Responses

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

secant squared theta minus tangent squared theta equals 1
Image with alt text: secant squared theta minus tangent squared theta equals 1

secant theta equals start fraction 1 over cosine theta end fraction
Image with alt text: secant theta equals start fraction 1 over cosine theta end fraction

cotangent theta equals start fraction 1 over tangent theta end fraction
Image with alt text: cotangent theta equals start fraction 1 over tangent theta end fraction

1 equals cosecant squared theta minus cotangent squared theta

Answer 1

To simplify the expression \(\frac{\sec^2 \theta - 1}{\cot^2 \theta + 1}\), Julia can use the following identities:

1. secant squared theta minus tangent squared theta equals 1: This identity can be rearranged to express \(\sec^2 \theta\) in terms of \(\tan^2 \theta\), which can be helpful since \(\sec^2 \theta = 1 + \tan^2 \theta\). Therefore, \(\sec^2 \theta - 1 = \tan^2 \theta\), which simplifies the numerator.

2. 1 equals cosecant squared theta minus cotangent squared theta: This identity can be rearranged to provide a relationship involving \(\cot^2 \theta\). Specifically, \(\cot^2 \theta = \csc^2 \theta - 1\), which can be useful for the denominator.

The first identity (sine squared theta plus cosine squared theta equals 1) and the third identity (secant theta equals \(\frac{1}{\cos \theta}\)) do not directly aid in simplifying this expression.

Thus, the identities Julia should use are:

• secant squared theta minus tangent squared theta equals 1
• 1 equals cosecant squared theta minus cotangent squared theta