## $$\color{Blue}{\textbf{Calculus}}$$

### $$\color{Magenta}{\textbf{1.Trigonometry}}$$

$\textbf{Pythagorean Identities:}$

• $\sin^{2}\theta + \cos^{2}\theta = 1$
• $1+\tan^{2}\theta = \sec^{2}\theta$
• $1+\cot^{2}\theta = \csc^{2}\theta$

$\textbf{Reciprocal Identities:}$

• $\sin\theta = \dfrac{1}{\csc\theta}$
• $\cos\theta = \dfrac{1}{\sec\theta}$
• $\tan\theta = \dfrac{1}{\cot\theta}$
• $\csc\theta = \dfrac{1}{\sin\theta}$
• $\sec\theta = \dfrac{1}{\cos\theta}$
• $\cot\theta = \dfrac{1}{\tan\theta}$

$\textbf{Tangent and Cotangent Identities:}$

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
• $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

$\textbf{Formulas for twice of angle:}$

• $\sin2\theta = 2 \sin\theta \cos\theta$
• $\cos2\theta = 2\cos^{2}\theta\: – 1 = 1 – 2\sin^{2}\theta$
• $\tan2\theta = \dfrac{2\tan\theta} {1− \tan^{2}\theta} = \dfrac{\sin 2\theta} {1− 2\sin^{2}\theta}$

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$\color{\Purple}{\textbf{Some important things, we should know}}$

• ${\color{Red} {\cos 0 = 1,\cos \pi = -1,\cos 2\pi = 1,\dots}}$
• ${\color{Blue}{ \text{In general}\: \cos n\pi = (-1)^{n}\: \text{where}\: n=0,1,2,\dots}}$
• ${\color{Magenta} {\sin 0 = 0,\sin \pi = 0,\sin 2\pi = 0,\dots}}$
• ${\color{Green} {\text{In general}\: \sin n\pi = 0\: \text{where}\: n=0,1,2,\dots}}$
• ${\color{Orange} {\sin(\pi-x)=\sin x,\sin (2\pi-x)=-\sin x,\sin(3\pi-x)=\sin x}}$
• ${\color{Teal} {\text{In general}\: \sin (n\pi-x)=(-1)^{n+1}\sin x\: \text{ where} \: n=0,1,2,\dots}}$
•  ${\color{Orchid} {\cos(\pi-x)=-\cos x,\cos (2\pi-x)=\cos x,\cos (3\pi-x)=-\cos x}}$
• ${\color{purple} {\text{ In general}\: \cos (n\pi-x)=(-1)^{n}\cos x \: \text{where}\: n=0,1,2,\dots}}$

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$f(x) = \sin x$

$f(x) = \cos x$

Visualization:

### $$\color{Red}{\textbf{2.Integral & Differential Calculus}}$$

$${\color{Teal}{2.1.\textbf{Some Useful Formulae for Differentiation}}}$$

1. $\dfrac{\mathrm{d} }{\mathrm{d} x}(c) = 0;c = \text{Constant}$
2. $\dfrac{\mathrm{d} }{\mathrm{d} x}(x) = 1$
3. $\dfrac{\mathrm{d} }{\mathrm{d} x}(x^{n}) = nx^{n-1}$
4. $\dfrac{\mathrm{d} }{\mathrm{d} x}(e^{x}) = e^{x}$
5. $\dfrac{\mathrm{d} }{\mathrm{d} x}(a^{x}) = a^{x}\log_{e}a$
6. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log x) = \dfrac{1}{x}$
7. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_{a} x) = \dfrac{1}{x}\log_{a}e$
8. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin x) = \cos x$
9. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos x) = -\sin x$
10. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan x) = \sec^{2} x$
11. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot x) = -\csc^{2} x$
12. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc x) = \csc x \cot x$
13. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec x) = \sec x\tan x$
14. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin^{-1} x) = \dfrac{1}{\sqrt{1-x^{2}}}$
15. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos^{-1} x) = \dfrac{-1}{\sqrt{1-x^{2}}}$
16. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan^{-1} x) = \dfrac{1}{1 + x^{2}}$
17. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot^{-1} x) = \dfrac{-1}{1 + x^{2}}$
18. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec^{-1} x) = \dfrac{1}{x \sqrt{x^{2}-1 }}$
19. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc^{-1} x) = \dfrac{-1}{x \sqrt{x^{2}-1}}$
20. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sqrt{x}) = \dfrac{1}{ 2 \sqrt{x}}$

$${\color{Purple}{2.2.\textbf{Some Useful Formulae for Integration}}}$$

$\textbf{Algebraic Functions}$

1. $\displaystyle{}\int k\: dx = kx + C$
2. $\displaystyle{}\int x^{n}\: dx = \dfrac{x^{n+1}}{n+1} + C$
3. $\displaystyle{}\int (ax + b)^{n}\: dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + C\:;(\text{for}\: n \neq -1)$
4. $\displaystyle{}\int \dfrac{1}{x}\: dx = \ln x + C\:;\text{(for positive values of$x$only)}$
5. $\displaystyle{}\int \dfrac{c}{ax + b}\: dx = \dfrac{c}{a}\ln(ax +b) + C$

$\textbf{Exponential Functions}$

1. $\displaystyle{}\int e^{x} dx = e^{x}+ C$
2. $\displaystyle{}\int a^{x} dx = \dfrac{a^{x}}{\ln a}+ C$

$\textbf{Logarithm Functions}$

1. $\displaystyle{}\int \ln x\: dx = x\ln x\: – x+ C$
2. $\displaystyle{}\int \log_{a} x\: dx = x\log_{a} x\: – \dfrac{x}{\log a} + C$

$\textbf{Trigonometric Functions}$

$\displaystyle{}\int \sin x\: dx = -\cos x + C$

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This post is not completed yet, I'll update all points soon.
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Sir, please do it as early as possible.
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@kashyap02

I'm not sir, I'm also an aspirant like yours.

I will update soon, after the GATE exam. It will very helpful for ISI, CMI exams.