Answer:
(identity has been verified)
Step-by-step 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)
