Question

Beginning with the key trigonometric identities, develop an identity for tan(2A).

Answer 1

The identity for tan(2A) can be derived from the key trigonometric identities.

Step-by-step explanation:

The identity for tan(2A) can be derived from the key trigonometric identities. Let's start with the double angle identity for sine: sin(2A) = 2sin(A)cos(A). We can rewrite this as cos(A) = sin(A)/cos(A). Now, let's substitute this into the definition of tan(A) = sin(A)/cos(A), which gives us:

tan(2A) = sin(2A)/cos(2A) = 2sin(A)cos(A)/[cos(A)]^2 = 2sin(A)cos(A)/[1 - sin^2(A)].

So, the identity for tan(2A) is:

tan(2A) = 2sin(A)cos(A)/[1 - sin^2(A)].