Verify the Identity (1-cos(A)^2)(1+cot(A)^2)=1

Problem

(1−cos^2(A))*(1+cot^2(A))=1

Solution

1. Identify the Pythagorean identity for the first factor, which states that sin^2(A)+cos^2(A)=1

2. Substitute 1−cos^2(A) with sin^2(A) in the expression.

1−cos^2(A)=sin^2(A)

3. Identify the Pythagorean identity for the second factor, which states that 1+cot^2(A)=csc^2(A)

4. Substitute 1+cot^2(A) with csc^2(A) in the expression.

1+cot^2(A)=csc^2(A)

5. Rewrite the expression using the substitutions made in the previous steps.

sin^2(A)*csc^2(A)

6. Apply the reciprocal identity csc(A)=1/sin(A) which implies csc^2(A)=1/sin^2(A)

sin^2(A)⋅1/sin^2(A)

7. Simplify the product by canceling the common terms.

sin^2(A)/sin^2(A)=1

Final Answer

(1−cos^2(A))*(1+cot^2(A))=1

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