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

Problem

(1−cos2(A))*(1+cot2(A))=1

Solution

1. Identify the Pythagorean identity for the first factor, which states that sin2(A)+cos2(A)=1

2. Substitute 1−cos2(A) with sin2(A) in the expression.

1−cos2(A)=sin2(A)

3. Identify the Pythagorean identity for the second factor, which states that 1+cot2(A)=csc2(A)

4. Substitute 1+cot2(A) with csc2(A) in the expression.

1+cot2(A)=csc2(A)

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

sin2(A)*csc2(A)

6. Apply the reciprocal identity csc(A)=1/sin(A) which implies csc2(A)=1/sin2(A)

sin2(A)⋅1/sin2(A)

7. Simplify the product by canceling the common terms.

sin2(A)/sin2(A)=1

Final Answer

(1−cos2(A))*(1+cot2(A))=1

Want more problems? Check here!