Question

Prove that, tan θ ( 1 + cot ^ 2 θ ) / ( 1 + tan ^ 2 θ ) = cot θ

Answer 1

(identity has been verified)

Explanation:

Verify the following identity:

tan(θ) (cot(θ)^2 + 1)/(tan(θ)^2 + 1) = cot(θ)

Multiply both sides by tan(θ)^2 + 1:

tan(θ) (cot(θ)^2 + 1) = ^?cot(θ) (tan(θ)^2 + 1)

(cot(θ)^2 + 1) tan(θ) = tan(θ) + cot(θ)^2 tan(θ):

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) (tan(θ)^2 + 1)

cot(θ) (tan(θ)^2 + 1) = cot(θ) + cot(θ) tan(θ)^2:

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) + cot(θ) tan(θ)^2

Write cotangent as cosine/sine and tangent as sine/cosine:

sin(θ)/cos(θ) + sin(θ)/cos(θ) (cos(θ)/sin(θ))^2 = ^?cos(θ)/sin(θ) + cos(θ)/sin(θ) (sin(θ)/cos(θ))^2

(sin(θ)/cos(θ)) + (cos(θ)/sin(θ))^2 (sin(θ)/cos(θ)) = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2

(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2 = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?cos(θ)/sin(θ) + sin(θ)/cos(θ)

The left hand side and right hand side are identical:

Answer: (identity has been verified)