Question

Cot^2 (x) tan^2 (x)=1

Prove the identity

Answer 1

To prove the identity cot²(x) tan²(x) = 1, we utilize the definitions of cotangent and tangent. By rewriting cot(x) as 1/tan(x), and substituting into the identity, we find that the expression simplifies to 1, confirming the identity as true.

Step-by-step explanation:

The question asks to prove the trigonometric identity cot²(x) tan²(x) = 1. Let's recall two fundamental trigonometric identities: tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x). By definition, cotangent is the reciprocal of tangent, or cot(x) = 1/tan(x).

We can rewrite the left side of the identity as:
(cos²(x)/sin²(x)) × (sin²(x)/cos²(x)),
which simplifies to cos²(x)sin²(x) / (sin²(x)cos²(x)). Since the numerator and denominator are the same, this simplifies to 1, which proves cot²(x) tan²(x) = 1 is indeed a true identity.