Trigonometric Identities: The Complete List with Examples

Trigonometry runs through geometry, calculus, and physics, but the sheer number of identities makes them easy to mix up. This guide collects every core group of trigonometric identities — from the Pythagorean and cofunction relations to the angle sum, double-angle, power-reduction, and product-to-sum formulas — with a memory tip for each group.

The fundamental identities

These are the foundation. Memorize them before anything else, because every later group is built from them:

$${\sin }^{2}x+{\cos }^{2}x=1$$

$$\tan x=\frac{\sin x}{\cos x}\cot x=\frac{\cos x}{\sin x}\tan x\cdot \cot x=1$$

$$1+{\tan }^{2}x=\frac{1}{{\cos }^{2}x}1+{\cot }^{2}x=\frac{1}{{\sin }^{2}x}$$

Conditions: $\cos x\ne 0$ whenever you use $\tan x$, and $\sin x\ne 0$ whenever you use $\cot x$.

Memory tip: the last two are just consequences of ${\sin }^{2}x+{\cos }^{2}x=1$. Divide both sides by ${\cos }^{2}x$ to get the $\tan$ identity, or by ${\sin }^{2}x$ to get the $\cot$ identity. (Note $\frac{1}{{\cos }^{2}x}={\sec }^{2}x$ and $\frac{1}{{\sin }^{2}x}={\csc }^{2}x$.)

Related-angle identities

Certain angles reduce back to the same reference angle, so you can rewrite them with a sign change:

Memory tip: cosine is even, so $\cos (-x)=\cos x$; sine and tangent are odd, so they pick up a minus sign. The complementary (cofunction) row is the key one: over a $\frac{\pi }{2}$ shift, sine and cosine swap. That is exactly why they are called cofunctions.

Angle sum and difference formulas

$$\sin (a\pm b)=\sin a\cos b\pm \cos a\sin b$$

$$\cos (a\pm b)=\cos a\cos b\mp \sin a\sin b$$

$$\tan (a\pm b)=\frac{\tan a\pm \tan b}{1\mp \tan a\tan b}$$

Memory tip: for $\sin$, it is "sine cosine, cosine sine" and the sign is kept the same. For $\cos$, it is "cosine cosine, sine sine" but the sign is flipped (so $\cos (a+b)$ takes a minus). The classic chant: "sine is sin-cos plus cos-sin; cosine is cos-cos minus sin-sin."

Double-angle formulas

Setting $b=a$ in the sum formulas gives the double-angle identities:

$$\sin 2a=2\sin a\cos a$$

$$\cos 2a={\cos }^{2}a-{\sin }^{2}a=2{\cos }^{2}a-1=1-2{\sin }^{2}a$$

$$\tan 2a=\frac{2\tan a}{1-{\tan }^{2}a}$$

Worked example. Given $\sin a=\frac{3}{5}$ with $0<a<\frac{\pi }{2}$, find $\sin 2a$ and $\cos 2a$.

Since $a$ is in the first quadrant, $\cos a>0$:

$$\cos a=\sqrt{1-{\sin }^{2}a}=\sqrt{1-\frac{9}{25}}=\frac{4}{5}$$

Therefore:

$$\sin 2a=2\cdot \frac{3}{5}\cdot \frac{4}{5}=\frac{24}{25},\cos 2a=1-2{\sin }^{2}a=1-\frac{18}{25}=\frac{7}{25}$$

Power-reduction formulas

These follow directly from the $\cos 2a$ identity and are used to turn a squared trig function into a first-degree one:

$${\sin }^{2}a=\frac{1-\cos 2a}{2}{\cos }^{2}a=\frac{1+\cos 2a}{2}$$

$${\tan }^{2}a=\frac{1-\cos 2a}{1+\cos 2a}$$

This group is invaluable when finding antiderivatives and solving trigonometric equations, because it replaces ${\sin }^2$ and ${\cos }^2$ with a linear expression in $\cos 2a$.

Product-to-sum formulas

$$\cos a\cos b=\frac{1}{2}[\cos (a-b)+\cos (a+b)]$$

$$\sin a\sin b=\frac{1}{2}[\cos (a-b)-\cos (a+b)]$$

$$\sin a\cos b=\frac{1}{2}[\sin (a-b)+\sin (a+b)]$$

Sum-to-product formulas

$$\cos a+\cos b=2\cos \frac{a+b}{2}\cos \frac{a-b}{2}$$

$$\cos a-\cos b=-2\sin \frac{a+b}{2}\sin \frac{a-b}{2}$$

$$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2}$$

$$\sin a-\sin b=2\cos \frac{a+b}{2}\sin \frac{a-b}{2}$$

Memory tip: every sum-to-product formula uses the half-sum $\frac{a+b}{2}$ and the half-difference $\frac{a-b}{2}$, with a factor of $2$ out front. Remember "half the sum, half the difference" and you only have to place the signs.

Common mistakes

• Wrong sign in $\cos (a+b)$: the cosine sum formula flips the sign, so $\cos (a+b)=\cos a\cos b-\sin a\sin b$ (minus, not plus).
• Forgetting the domain restrictions: $\tan$ requires $\cos x\ne 0$, and $\cot$ requires $\sin x\ne 0$.
• Confusing complementary and supplementary angles: complementary adds to $\frac{\pi }{2}$ and swaps $\sin \leftrightarrow \cos$; supplementary adds to $\pi$, keeps $\sin$, and flips the sign of $\cos$.
• Using only one form of $\cos 2a$: all three forms are correct — pick the one that matches the value the problem hands you (as in the worked example, where $1-2{\sin }^{2}a$ was fastest).

Master these seven groups and you have almost the entire toolkit for solving trig equations and simplifying expressions. Try entering the examples above into the calculator to check each step of your own work.