Final answer:
To prove that the trigonometric expression 1 - cos2A ÷ Sin2A is equal to tanA, we use trigonometric identities such as the Pythagorean identity and the double angle formula. This simplifies the expression down to tanA, confirming the equality.
Step-by-step explanation:
To prove that 1 - cos2A ÷ Sin2A equals tanA, we start by using trigonometric identities to rewrite the original expression:
- Replace 1 - cos2A with sin2A using the Pythagorean identity: cos2θ + sin2θ = 1.
- Rewrite Sin2A as 2sinAcosA using the double angle formula: sin 2θ = 2sinθ cosθ.
This gives us:
(1 - cos2A) ÷ Sin2A = sin2A ÷ (2sinAcosA)
When we simplify the expression by canceling out sinA from the numerator and denominator, we are left with 1÷(2cosA), which simplifies further to sinA÷cosA, and we know that sinA÷cosA is the definition of tanA. Thus, the initial expression is indeed equal to tanA.