Trig :(

Every important trig thing i could think of, except for trig sub, ill add that in a few weeks.

Double-Angle Identity for Sine

$$\sin(2x) = 2\sin(x)\cos(x)$$

Double-Angle Identity for Cosine

$$\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$$

Double-Angle Identity for Tangent

$$\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$$

Double-Angle Identity for Cotangent

$$\cot(2x) = \frac{\cot^2(x) - 1}{2\cot(x)}$$

Half-Angle Identity for Sine

$$\sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos(x)}{2}}$$

Half-Angle Identity for Cosine

$$\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{2}}$$

Half-Angle Identity for Tangent

$$\tan\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} = \frac{1 - \cos(x)}{\sin(x)} = \frac{\sin(x)}{1 + \cos(x)} = \csc(x) - \cot(x)$$

Half-Angle Identity for Cotangent

$$\cot\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{1 - \cos(x)}} = \frac{1 + \cos(x)}{\sin(x)} = \frac{\sin(x)}{1 - \cos(x)} = \csc(x) + \cot(x)$$

Squared Half-Angle Identity for Sine

$$\sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{2}$$

Squared Half-Angle Identity for Cosine

$$\cos^2\left(\frac{x}{2}\right) = \frac{1 + \cos(x)}{2}$$

Squared Half-Angle Identity for Tangent

$$\tan^2\left(\frac{x}{2}\right) = \frac{1 - \cos(x)}{1 + \cos(x)}$$

Power-Reduction Formula for $\sin^2(x)$

$$\sin^2(x) = \frac{1 - \cos(2x)}{2}$$

Power-Reduction Formula for $\cos^2(x)$

$$\cos^2(x) = \frac{1 + \cos(2x)}{2}$$

Power-Reduction Formula for $\tan^2(x)$

$$\tan^2(x) = \frac{1 - \cos(2x)}{1 + \cos(2x)} = \sec^2(x) - 1$$

Power-Reduction Formula for $\sin^3(x)$

$$\sin^3(x) = \frac{3\sin(x) - \sin(3x)}{4}$$

Power-Reduction Formula for $\cos^3(x)$

$$\cos^3(x) = \frac{3\cos(x) + \cos(3x)}{4}$$

Power-Reduction Formula for $\sin^4(x)$

$$\sin^4(x) = \frac{3 - 4\cos(2x) + \cos(4x)}{8}$$

Power-Reduction Formula for $\cos^4(x)$

$$\cos^4(x) = \frac{3 + 4\cos(2x) + \cos(4x)}{8}$$

Power-Reduction Formula for $\sin^2(x)\cos^2(x)$

$$\sin^2(x)\cos^2(x) = \frac{1 - \cos(4x)}{8}$$

Product-to-Sum Identity for $\sin(A)\cos(B)$

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

Product-to-Sum Identity for $\cos(A)\sin(B)$

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

Product-to-Sum Identity for $\cos(A)\cos(B)$

$$\cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$$

Product-to-Sum Identity for $\sin(A)\sin(B)$

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

Sum-to-Product Identity for $\sin(A) + \sin(B)$

$$\sin(A) + \sin(B) = 2\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$$

Sum-to-Product Identity for $\sin(A) - \sin(B)$

$$\sin(A) - \sin(B) = 2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$$

Sum-to-Product Identity for $\cos(A) + \cos(B)$

$$\cos(A) + \cos(B) = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$$

Sum-to-Product Identity for $\cos(A) - \cos(B)$

$$\cos(A) - \cos(B) = -2\sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$$

Angle Sum and Difference Identity for Sine

$$\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$$

Angle Sum and Difference Identity for Cosine

$$\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$$

Angle Sum and Difference Identity for Tangent

$$\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$$

Pythagorean Identity for Sine and Cosine

$$\sin^2(x) + \cos^2(x) = 1$$

Pythagorean Identity for Tangent and Secant

$$1 + \tan^2(x) = \sec^2(x)$$

Pythagorean Identity for Cotangent and Cosecant

$$1 + \cot^2(x) = \csc^2(x)$$

Reciprocal Identity for Cosecant

$$\csc(x) = \frac{1}{\sin(x)}$$

Reciprocal Identity for Secant

$$\sec(x) = \frac{1}{\cos(x)}$$

Reciprocal Identity for Cotangent

$$\cot(x) = \frac{1}{\tan(x)}$$

Derivative of $\sin(x)$

$$\frac{d}{dx}[\sin(x)] = \cos(x)$$

Derivative of $\cos(x)$

$$\frac{d}{dx}[\cos(x)] = -\sin(x)$$

Derivative of $\tan(x)$

$$\frac{d}{dx}[\tan(x)] = \sec^2(x)$$

Derivative of $\cot(x)$

$$\frac{d}{dx}[\cot(x)] = -\csc^2(x)$$

Derivative of $\sec(x)$

$$\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$$

Derivative of $\csc(x)$

$$\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$$

Chain Rule Derivative of $\sin(u)$

$$\frac{d}{dx}[\sin(u)] = \cos(u)u'$$

Chain Rule Derivative of $\tan(u)$

$$\frac{d}{dx}[\tan(u)] = \sec^2(u)u'$$

Integral of $\sin(x)$

$$\int \sin(x)\,dx = -\cos(x) + C$$

Integral of $\cos(x)$

$$\int \cos(x)\,dx = \sin(x) + C$$

Integral of $\tan(x)$

$$\int \tan(x)\,dx = -\ln(|\cos(x)|) + C = \ln(|\sec(x)|) + C$$

Integral of $\cot(x)$

$$\int \cot(x)\,dx = \ln(|\sin(x)|) + C = -\ln(|\csc(x)|) + C$$

Integral of $\sec(x)$

$$\int \sec(x)\,dx = \ln(|\sec(x) + \tan(x)|) + C$$

Integral of $\csc(x)$

$$\int \csc(x)\,dx = -\ln(|\csc(x) + \cot(x)|) + C = \ln(|\csc(x) - \cot(x)|) + C$$

Integral of $\sec^2(x)$

$$\int \sec^2(x)\,dx = \tan(x) + C$$

Integral of $\sec(x)\tan(x)$

$$\int \sec(x)\tan(x)\,dx = \sec(x) + C$$

Integral of $\csc^2(x)$

$$\int \csc^2(x)\,dx = -\cot(x) + C$$

Integral of $\csc(x)\cot(x)$

$$\int \csc(x)\cot(x)\,dx = -\csc(x) + C$$

Integral of $\sin(ax + b)$

$$\int \sin(ax + b)\,dx = -\frac{1}{a}\cos(ax + b) + C$$

Integral of $\cos(ax + b)$

$$\int \cos(ax + b)\,dx = \frac{1}{a}\sin(ax + b) + C$$

Integral of $\sec^2(ax + b)$

$$\int \sec^2(ax + b)\,dx = \frac{1}{a}\tan(ax + b) + C$$

Derivative of $\arcsin(x)$

$$\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}$$

Derivative of $\arccos(x)$

$$\frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}$$

Derivative of $\arctan(x)$

$$\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$$

Derivative of $\operatorname{arccot}(x)$

$$\frac{d}{dx}[\operatorname{arccot}(x)] = -\frac{1}{1 + x^2}$$

Derivative of $\operatorname{arcsec}(x)$

$$\frac{d}{dx}[\operatorname{arcsec}(x)] = \frac{1}{|x|\sqrt{x^2 - 1}}$$

Derivative of $\operatorname{arccsc}(x)$

$$\frac{d}{dx}[\operatorname{arccsc}(x)] = -\frac{1}{|x|\sqrt{x^2 - 1}}$$

Standard Integral Form for $\arcsin(x)$

$$\int \frac{dx}{\sqrt{1 - x^2}} = \arcsin(x) + C$$

Standard Integral Form for $\arctan(x)$

$$\int \frac{dx}{1 + x^2} = \arctan(x) + C$$

Generalized Integral Form for $\arcsin\left(\frac{x}{a}\right)$

$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C$$

Generalized Integral Form for $\arctan\left(\frac{x}{a}\right)$

$$\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$$

Generalized Integral Form for $\operatorname{arcsec}\left|\frac{x}{a}\right|$

$$\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a}\operatorname{arcsec}\left|\frac{x}{a}\right| + C$$