Question

Simplify and write the trigonometric expression in terms of sine and cosine: 2 tan 2 x sec 2 x − 1 = g ( x )

a) g(x) = sin(2x)
b) g(x) = cos(2x)
c) g(x) = sin²(x)
d) g(x) = cos²(x)

Answer 1

Using trigonometric identities, the given expression simplifies to g(x) = sin(2x), which corresponds to option a).

Step-by-step explanation:

Simplifying the trigonometric expression 2 tan 2x sec 2x - 1 and writing it in terms of sine and cosine involves using trigonometric identities. Let's start by remembering that tan θ = sin θ / cos θ. Therefore, tan 2x = sin 2x / cos 2x. Also, sec θ = 1 / cos θ, which means sec 2x = 1 / cos 2x. Multiplying these together gives 2 tan 2x sec 2x = 2 (sin 2x / cos 2x) (1 / cos 2x) or 2 sin 2x / cos^2 2x. When we multiply this expression by 2, we are left with sin 2x, due to the Pythagorean identity cos2 θ = 1 - sin2 θ.

The expression simplifies to sin 2x, as the secant squared and cosine squared terms cancel out, fulfilling the Pythagorean identity. Therefore, g(x) = sin(2x), which corresponds to option a).