Worked example

Simplify sin²θ + cos²θ + 3cos²θ

Answer

1 + 3cos²θ — and with the power-reducing formula, 5/2 + (3/2)cos 2θ.

sin²θ + cos²θ + 3cos²θ
Original expression.
= 1 + 3cos²θ
The first two terms are the Pythagorean identity. · Pythagorean identity
= 1 + 3(1 + cos 2θ)/2
Power-reducing formula for cos²θ. · Power-reducing formula for cos²
= 5/2 + (3/2)cos 2θ
Collect constants.

sin²θ + cos²θ = 1

cos²x = (1 + cos 2x)/2

Pythagorean identity — Two squared trig terms of the same angle are added: the pair is exactly 1, so replace both with 1 — or, going the other way, replace a stubborn 1 with sin²θ + cos²θ to get a common denominator.
Power-reducing formula for cos² — The cosine twin: ∫cos²x dx, RMS values of a wave, and any average of cos² over a whole number of periods (the cosine term integrates to 0 and 1/2 is left).

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.