In this lesson you’ll learn the double-angle, power-reducing, and half-angle identities, and how to use them to simplify expressions and solve equations. You covered sum and difference identities in Trigonometry. Here we build on those to derive more powerful tools.
All of these identities come from the sum formulas you already know. The double-angle formulas are just the sum formulas with A = B.
Sum and difference identities (review):
sin ( A ± B ) = sin A cos B ± cos A sin B \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B sin ( A ± B ) = sin A cos B ± cos A sin B cos ( A ± B ) = cos A cos B ∓ sin A sin B \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B cos ( A ± B ) = cos A cos B ∓ sin A sin B Double-angle identities (set B = A in the sum formulas):
sin 2 θ = 2 sin θ cos θ \sin 2\theta = 2\sin\theta\cos\theta sin 2 θ = 2 sin θ cos θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ π 2π sin(x) sin(2x)
The graph shows how sin(2x) (orange) completes two full cycles in the same interval where sin(x) (blue) completes one. That’s the double-angle in action.
Power-reducing identities (solve the cos 2θ formulas for sin²θ and cos²θ):
sin 2 θ = 1 − cos 2 θ 2 , cos 2 θ = 1 + cos 2 θ 2 \sin^2\theta = \frac{1 - \cos 2\theta}{2}, \quad \cos^2\theta = \frac{1 + \cos 2\theta}{2} sin 2 θ = 2 1 − cos 2 θ , cos 2 θ = 2 1 + cos 2 θ Half-angle identities (replace θ with θ/2 in the power-reducing formulas):
sin θ 2 = ± 1 − cos θ 2 , cos θ 2 = ± 1 + cos θ 2 \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}, \quad \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}} sin 2 θ = ± 2 1 − cos θ , cos 2 θ = ± 2 1 + cos θ The ± depends on which quadrant θ/2 is in.
Example 1: Use double-angle to simplify
Simplify cos 2 θ − sin 2 θ \cos^2\theta - \sin^2\theta cos 2 θ − sin 2 θ
cos 2 θ − sin 2 θ = cos 2 θ \cos^2\theta - \sin^2\theta = \cos 2\theta cos 2 θ − sin 2 θ = cos 2 θ That’s it. Recognizing the pattern is the whole skill.
Example 2: Power-reducing
Express sin 2 3 θ \sin^2 3\theta sin 2 3 θ
sin 2 3 θ = 1 − cos 6 θ 2 \sin^2 3\theta = \frac{1 - \cos 6\theta}{2} sin 2 3 θ = 2 1 − cos 6 θ This is useful in calculus when integrating sin 2 ( 3 θ ) \sin^2(3\theta) sin 2 ( 3 θ )
Example 3: Solve using double-angle
Solve sin 2 x = cos x \sin 2x = \cos x sin 2 x = cos x [ 0 , 2 π ) [0, 2\pi) [ 0 , 2 π )
Replace sin 2x with the double-angle formula:
2 sin x cos x = cos x 2\sin x \cos x = \cos x 2 sin x cos x = cos x 2 sin x cos x − cos x = 0 2\sin x \cos x - \cos x = 0 2 sin x cos x − cos x = 0 cos x ( 2 sin x − 1 ) = 0 \cos x(2\sin x - 1) = 0 cos x ( 2 sin x − 1 ) = 0 So cos x = 0 \cos x = 0 cos x = 0 sin x = 1 / 2 \sin x = 1/2 sin x = 1/2
From cos x = 0 \cos x = 0 cos x = 0 x = π / 2 x = \pi/2 x = π /2 x = 3 π / 2 x = 3\pi/2 x = 3 π /2
From sin x = 1 / 2 \sin x = 1/2 sin x = 1/2 x = π / 6 x = \pi/6 x = π /6 x = 5 π / 6 x = 5\pi/6 x = 5 π /6
Four solutions: x = π / 6 , π / 2 , 5 π / 6 , 3 π / 2 x = \pi/6, \; \pi/2, \; 5\pi/6, \; 3\pi/2 x = π /6 , π /2 , 5 π /6 , 3 π /2
Example 4: Half-angle
Find the exact value of cos 15 ° \cos 15° cos 15°
cos 15 ° = cos 30 ° 2 = 1 + cos 30 ° 2 = 1 + 3 2 2 = 2 + 3 4 = 2 + 3 2 \cos 15° = \cos\frac{30°}{2} = \sqrt{\frac{1 + \cos 30°}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} cos 15° = cos 2 30° = 2 1 + cos 30° = 2 1 + 2 3 = 4 2 + 3 = 2 2 + 3 Advanced trig identities are used in:
Electrical engineering (simplifying AC circuit analysis, power calculations)
Physics (wave interference, standing waves, harmonic motion)
Signal processing (Fourier analysis breaks signals into sine/cosine components)
Computer graphics (rotation matrices, animation curves)
Calculus (integration of trig functions requires power-reducing identities)
Example: when two sound waves interfere, the resulting amplitude involves product-to-sum identities. Engineers use these to analyze beats and resonance.
Advanced trig identities are tools for simplifying complicated expressions. Start by memorizing the double-angle formulas, as they appear most often. When verifying identities, work on the more complicated side first. With practice, these become powerful shortcuts in both trigonometry and calculus.
The double-angle formula for sine is: A. $\sin 2\theta = \sin^2\theta - \cos^2\theta$ B. $\sin 2\theta = \cos\theta - \sin\theta$ C. $\sin 2\theta = 1 - 2\cos^2\theta$ D. $\sin 2\theta = 2\sin\theta\cos\theta$
$\cos 2\theta$ can be written as: A. $\cos^2\theta - \sin^2\theta$ B. $2\sin\theta\cos\theta$ C. $\sin\theta + \cos\theta$ D. $1 + \sin^2\theta$
The power-reducing formula for $\sin^2\theta$ is: A. $\frac{1 + \cos 2\theta}{2}$ B. $\frac{1 - \cos 2\theta}{2}$ C. $2\sin\theta\cos\theta$ D. $\cos 2\theta$
When verifying trig identities, it's usually best to: A. Work on the simpler side first B. Add both sides together C. Work on the more complicated side first D. Multiply both sides by the same term
Solving $\sin 2x = \cos x$ starts by replacing $\sin 2x$ with: A. $\cos^2 x - \sin^2 x$ B. $1 - 2\sin^2 x$ C. $\sin x + \cos x$ D. $2\sin x\cos x$
Retry Quiz
