Trigonometric Identity Proof with Justifications
The image contains two steps of a proof of a trigonometric identity, and there is a question asking which option best justifies the equation for each step.
Step 1: cos^2(θ) + [csc(θ) - sin(θ)]^2 = cos^2(θ) + [csc^2(θ) - 2csc(θ)sin(θ) + sin^2(θ)].
The justification for Step 1 is:
**Option C: Using a form of the Pythagorean Identity.**
This is because the term [csc(θ) - sin(θ)]^2 is expanded using the distributive property (a - b)^2 = a^2 - 2ab + b^2, and csc^2(θ) is the reciprocal identity of sin^2(θ) stated by the Pythagorean identity csc^2(θ) = 1/sin^2(θ).
Step 2: cos^2(θ) + [csc^2(θ) - 2csc(θ)sin(θ) + sin^2(θ)] = cos^2(θ) + [1 + sin^2(θ) - 2csc(θ)sin(θ)].
The justification for Step 2 is:
**Option B: Rearranging terms.**
The terms have been reorganized to group the 1 (which comes from the identity csc^2(θ) = 1/sin^2(θ)) with sin^2(θ) to show how they come together due to the use of the Pythagorean identity. In this case, the "Rearranging terms" option best describes the action taken to move the 1 next to sin^2(θ) to later apply the identity sin^2(θ) + 1 = csc^2(θ).
Please note that although the Pythagorean identity is used again in Step 2, the rearrangement of terms to show the identity is the focus of this step, which is why Option B is chosen over A.
