Trig Identities Calculator

Calculate and verify trigonometric identities step-by-step. Solve trig equations using Pythagorean identities, double angle formulas, half angle formulas, and sum and difference identities.

Identity Type:

Trigonometric Function:

sin

cos

tan

csc

sec

cot

Angle (θ in degrees):

Or enter your expression:

Trigonometric Identity Result

Original expression will appear here

Simplified result will appear here

Selected Function	sin
Angle (θ)	30°
Exact Value	1/2
Decimal Value	0.5

About Trig Identities Calculator

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are useful for simplifying trigonometric expressions, solving trigonometric equations, and proving other mathematical statements.

Pythagorean Identities

The fundamental Pythagorean identities are derived from the Pythagorean theorem and the unit circle:

\[ \sin^2\theta + \cos^2\theta = 1 \] \[ 1 + \tan^2\theta = \sec^2\theta \] \[ 1 + \cot^2\theta = \csc^2\theta \]

Double Angle Identities

Double angle identities express trigonometric functions of double angles (2θ) in terms of single angles (θ):

\[ \sin(2\theta) = 2\sin\theta\cos\theta \] \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \] \[ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} \]

Half Angle Identities

Half angle identities express trigonometric functions of half angles (θ/2) in terms of the original angle (θ):

\[ \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}} \] \[ \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}} \] \[ \tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} \]

Sum and Difference Identities

These identities express trigonometric functions of sums or differences of angles in terms of functions of the individual angles:

\[ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \] \[ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \] \[ \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta} \]

Product-to-Sum Identities

These identities convert products of trigonometric functions into sums or differences:

\[ \sin\alpha\sin\beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)] \] \[ \cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)] \] \[ \sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)] \]