Formulas

Rules of Indices

1. $$a^m×a^n=a^{m+n}$$
2. $$a^m÷a^n=a^{m-n}$$
3. $$(a^m)^n=a^{mn}$$
4. $$a^m×b^m=(a×b)^m$$
5. $$a^m÷b^m=\left(\frac{a}{b}\right)^m$$
6. $$a^0=1$$
7. $$a^{-n}=\frac{1}{a^n}$$

Algebraic Identities

1. $$(a+b)^2=a^2+2ab+b^2$$
2. $$(a-b)^2=a^2-2ab+b^2$$
3. $$(a+b)^3=a^3+3a^2 b+3ab^2+b^2$$
4. $$(a-b)^3=a^3-3a^2 b+3ab^2-b^2$$
5. $$a^2-b^2=(a-b)(a+b)$$
6. $$a^3-b^3=(a-b)(a^2+ab+b^2)$$
7. $$a^3+b^3=(a+b)(a^2-ab+b^2)$$

Logarithm

$$log_b \,a = c \iff b^c = a$$ $$\textup{where} \; b \neq 1, a \gt 0$$

Rules of Logarithm

1. $$log_a \,xy = log_a \,x + log_a \,y$$
2. $$log_a \,\frac{x}{y} = log_a \,x - log_a \,y$$
3. $$log_a \,x^n = n log_a \,x$$
4. $$log_a \,1 = 0$$
5. $$log_a \,a = 1$$
6. $$log_y \,x = \frac{log_a \,x}{log_a \,y}$$
7. $$log_y \,x = \frac{1}{log_x \,y}$$

Trigonometric Ratios

1. $$\sin\theta = \frac{opposite}{hypotenuse}$$
2. $$\cos\theta = \frac{adjacent}{hypotenuse}$$
3. $$\tan\theta = \frac{opposite}{adjacent}$$
4. $$\cot\theta = \frac{adjacent}{opposite}$$
5. $$\sec\theta = \frac{hypotenuse}{adjacent}$$
6. $$\textup{cosec}\,\theta = \frac{hypotenuse}{opposite}$$

Reciprocal of Trigonometric Ratios

1. $$\sin\theta = \frac{1}{\textup{cosec}\,\theta}$$
2. $$\cos\theta = \frac{1}{\sec\theta}$$
3. $$\tan\theta = \frac{1}{\cot\theta}$$
4. $$\cot\theta = \frac{1}{\tan\theta}$$
5. $$\sec\theta = \frac{1}{\cos\theta}$$
6. $$\textup{cosec}\,\theta = \frac{1}{\sin\theta}$$
7. $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$
8. $$\cot\theta = \frac{\cos\theta}{\sin\theta}$$

Trigonometric Identities

1. $$\cos^2\theta + \sin^2\theta = 1$$
2. $$1 + \tan^2\theta = \sec^2\theta$$
3. $$1 + \cot^2\theta = \textup{cosec}^2\theta$$

Trigonometric Functions of Negative Angles

1. $$\sin(-\theta) = -\sin\theta$$
2. $$\cos(-\theta) = \cos\theta $$
3. $$\tan(-\theta) = -\tan\theta$$
4. $$\cot(-\theta) = -\cot\theta$$
5. $$\sec(-\theta) = \sec\theta$$
6. $$\textup{cosec}\,(-\theta) = -\textup{cosec}\,\theta$$

Addition Formulas of Trigonometric Functions

1. $$\sin(\alpha+\beta) = \sin\alpha\cos\beta+\cos\alpha\sin\beta$$
2. $$\sin(\alpha-\beta) = \sin\alpha\cos\beta-\cos\alpha\sin\beta$$
3. $$\cos(\alpha+\beta) = \cos\alpha\cos\beta-\sin\alpha\sin\beta$$
4. $$\cos(\alpha-\beta) = \cos\alpha\cos\beta+\sin\alpha\sin\beta$$
5. $$\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$
6. $$\tan(\alpha-\beta) = \frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$$
7. $$\cot(\alpha+\beta) = \frac{\cot\alpha\cot\beta-1}{\cot\beta+\cot\alpha}$$
8. $$\cot(\alpha-\beta) = \frac{\cot\alpha\cot\beta+1}{\cot\beta-\cot\alpha}$$

Double Angle Formulas of Trigonometric Functions

1. $$\sin 2\theta = 2\sin\theta\cos\theta$$
2. $$\cos 2\theta = \cos^2\theta-\sin^2\theta $$
3. $$\cos 2\theta = 2\cos^2\theta-1$$
4. $$\cos 2\theta = 1-2\sin^2\theta $$
5. $$\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}$$

Half Angle Formulas of Trigonometric Functions

1. $$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{2}}$$
2. $$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos\theta}{2}}$$
3. $$\tan \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$$
4. $$\tan \frac{\theta}{2} = \frac{\sin\theta}{1+\cos\theta}$$
5. $$\tan \frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}$$
6. $$\tan \frac{\theta}{2} = \textup{cosec}\theta-\cot\theta$$

Sum, Difference and Product of Trigonometric Functions

1. $$\sin\alpha+\sin\beta = 2\sin\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2}$$
2. $$\sin\alpha-\sin\beta = 2\cos\frac{\alpha+\beta}{2} \sin\frac{\alpha-\beta}{2}$$
3. $$\cos\alpha+\cos\beta = 2\cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2}$$
4. $$\cos\alpha-\cos\beta = 2\sin\frac{\alpha+\beta}{2} \sin\frac{\alpha-\beta}{2}$$
5. $$\sin\alpha\sin\beta = \frac{1}{2}[\cos(\alpha-\beta) - \cos(\alpha+\beta)]$$
6. $$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]$$
7. $$\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]$$

Relationship Between Exponential and Trigonometric Functions

1. $$e^{i\theta} = \cos\theta + i\sin\theta$$
2. $$e^{-i\theta} = \cos\theta - i\sin\theta$$
3. $$\sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2i}$$
4. $$\cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}$$
5. $$\tan\theta = \frac{e^{i\theta}-e^{-i\theta}}{i(e^{i\theta}+e^{-i\theta})}$$
6. $$\cot\theta = \frac{i(e^{i\theta}+e^{-i\theta})}{e^{i\theta}-e^{-i\theta}}$$
7. $$\sec\theta = \frac{2}{e^{i\theta}-e^{-i\theta}}$$
8. $$\textup{cosec}\,\theta = \frac{2i}{e^{i\theta}-e^{-i\theta}}$$

Hyperbolic Functions

1. $$\sinh x = \frac{e^x-e^{-x}}{2}$$
2. $$\cosh x = \frac{e^x+e^{-x}}{2}$$
3. $$\tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}$$
4. $$\coth x = \frac{e^x+e^{-x}}{e^x-e^{-x}}$$
5. $$\textup{sech}\,x = \frac{2}{e^x+e^{-x}}$$
6. $$\textup{cosech}\,x = \frac{2}{e^x-e^{-x}}$$

Reciprocal of Hyperbolic Functions

1. $$\sinh\theta = \frac{1}{\textup{cosech}\,\theta}$$
2. $$\cosh\theta = \frac{1}{\textup{sech}\theta}$$
3. $$\tanh\theta = \frac{1}{\coth\theta}$$
4. $$\coth\theta = \frac{1}{\tanh\theta}$$
5. $$\textup{sech}\theta = \frac{1}{\cosh\theta}$$
6. $$\textup{cosech}\,\theta = \frac{1}{\sinh\theta}$$
7. $$\tanh\theta = \frac{\sinh\theta}{\cosh\theta}$$
8. $$\coth\theta = \frac{\cosh\theta}{\sinh\theta}$$

Hyperbolic Identities

1. $$\cosh^2\theta - \sin^2\theta = 1$$
2. $$1 - \tanh^2\theta = \textup{sech}^2\theta$$
3. $$\coth^2\theta - 1 = \textup{cosech}^2\theta$$

Hyperbolic Functions of Negative Arguments

1. $$\sinh(-x) = -\sinh x$$
2. $$\cosh(-x) = \cosh x$$
3. $$\tanh(-x) = -\tanh x$$
4. $$\coth(-x) = -\coth x$$
5. $$\textup{sech}(-x) = \textup{sech}\, x$$
6. $$\textup{cosech}\,(-x) = -\textup{cosech}\, x$$

Addition Formulas of Hyperbolic Functions

1. $$\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y$$
2. $$\sinh(x-y) = \sinh x \cosh y - \cosh x \sinh y$$
3. $$\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y$$
4. $$\cosh(x-y) = \cosh x \cosh y + \sinh x\sinh y$$
5. $$\tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}$$
6. $$\tanh(x-y) = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y}$$
7. $$\coth(x+y) = \frac{\coth x \coth y + 1}{\coth y + \coth x}$$
8. $$\coth(x-y) = \frac{\coth x \coth y - 1}{\coth y - \coth x}$$

Double Angle Formulas of Hyperbolic Functions

1. $$\sinh 2x = 2\sinh x \cosh x$$
2. $$\cosh 2x = \cosh^2 x + \sinh^2 x $$
3. $$\cosh 2x = 2\cosh^2 x - 1$$
4. $$\cosh 2x = 1 + 2\sinh^2 x $$
5. $$\tanh 2x = \frac{2\tanh x}{1+\tanh^2 x}$$

Half Angle Formulas of Hyperbolic Functions

1. $$\sinh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{2}}$$
2. $$\cosh \frac{x}{2} = \sqrt{\frac{\cosh x + 1}{2}}$$
3. $$\tanh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{\cosh x + 1}}$$
4. $$\tanh \frac{x}{2} = \frac{\sinh x}{1 + \cosh x}$$
5. $$\tanh \frac{x}{2} = \frac{\cosh x - 1}{\sinh x}$$

Sum, Difference and Product of Trigonometric Functions

1. $$\sinh x + \sinh y = 2\sinh\frac{x+y}{2} \cosh\frac{x-y}{2}$$
2. $$\sinh x - \sinh y = 2\cosh\frac{x+y}{2} \sinh\frac{x-y}{2}$$
3. $$\cosh x + \cosh y = 2\cosh\frac{x+y}{2} \cosh\frac{x-y}{2}$$
4. $$\cosh x - \cosh y = 2\sinh\frac{x+y}{2} \sinh\frac{x-y}{2}$$
5. $$\sinh x \sinh y = \frac{1}{2}[\cosh(x-y) - \cosh(x+y)]$$
6. $$\cosh x\cosh y = \frac{1}{2}[\cosh(x-y) + \cosh(x+y)]$$
7. $$\sinh x\cosh y = \frac{1}{2}[\sinh(x+y) + \sinh(x-y)]$$

Relation between Inverse Trigonometric Functions

1. $$\sin^{-1}x+\cos^{-1}x = \frac{\pi}{2}$$
2. $$\tan^{-1}x+\cot^{-1}x = \frac{\pi}{2}$$
3. $$\sec^{-1}x+\textup{cosec}^{-1}x = \frac{\pi}{2}$$
4. $$\textup{cosec}^{-1}x = \sin^{-1}(1/x)$$
5. $$\sec^{-1}x = \cos^{-1}(1/x)$$
6. $$\cot^{-1}x = \tan^{-1}(1/x)$$
7. $$\sin^{-1}(-x) = -\sin^{-1}x$$
8. $$\cos^{-1}(-x) = \pi-\cos^{-1}x$$
9. $$\tan^{-1}(-x) = -\tan^{-1}x$$
10. $$\cot^{-1}(-x) = \pi-\cot^{-1}x$$
11. $$\sec^{-1}(-x) = \pi-\sec^{-1}x$$
12. $$\textup{cosec}^{-1}(-x) = -\textup{cosec}^{-1}x$$

Differentiation

1. $$\frac{d}{dx}x^n = nx^{n-1}$$
2. $$\frac{d}{dx}c = 0$$
3. $$\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}$$
4. $$\frac{d}{dx}e^x = e^x$$
5. $$\frac{d}{dx}a^x = a^x\,ln\,x$$

Derivatives of Trigonometric Functions

1. $$\frac{d}{dx}\sin x = \cos x$$
2. $$\frac{d}{dx}\cos x = -\sin x$$
3. $$\frac{d}{dx}\tan x = \sec^2 x$$
4. $$\frac{d}{dx}\cot x =-\textup{cosec}^2 x$$
5. $$\frac{d}{dx}\sec x = \sec x \tan x$$
6. $$\frac{d}{dx}\textup{cosec} x = -\textup{cosec} x \cot x$$

Derivatives of Inverse Trigonometric Functions

1. $$\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1-x^2}}$$
2. $$\frac{d}{dx}\cos^{-1} x = -\frac{1}{\sqrt{1-x^2}}$$
3. $$\frac{d}{dx}\tan^{-1} x = \frac{1}{1+x^2}$$
4. $$\frac{d}{dx}\cot^{-1} x = -\frac{1}{1+x^2}$$
5. $$\frac{d}{dx}\sec^{-1} x = \frac{1}{x \sqrt{x^2-1}}$$
6. $$\frac{d}{dx}\textup{cosec}^{-1} x = -\frac{1}{x \sqrt{x^2-1}}$$

Derivatives of Hyperbolic Functions

1. $$\frac{d}{dx}\sinh x = \cosh x$$
2. $$\frac{d}{dx}\cosh x = \sinh x$$
3. $$\frac{d}{dx}\tanh x = \textup{sech}^2 x$$
4. $$\frac{d}{dx}\coth x = -\textup{cosech}^2 x$$
5. $$\frac{d}{dx}\textup{sech} x = -\textup{sech} x \tanh x$$
6. $$\frac{d}{dx}\textup{cosech} x = -\textup{cosech} x \coth x$$

Derivatives of Inverse Hyperbolic Functions

1. $$\frac{d}{dx}\sinh^{-1} x = \frac{1}{\sqrt{x^2+1}}$$
2. $$\frac{d}{dx}\cosh^{-1} x = \frac{\pm 1}{\sqrt{x^2+1}}$$
3. $$\frac{d}{dx}\tanh^{-1} x = \frac{1}{1-x^2}$$
4. $$\frac{d}{dx}\coth^{-1} x = \frac{1}{1-x^2}$$
5. $$\frac{d}{dx}\textup{sech}^{-1} x = \frac{\pm 1}{x \sqrt{1-x^2}}$$
6. $$\frac{d}{dx}\textup{cosech}^{-1} x = \frac{\pm 1}{x \sqrt{1+x^2}}$$