Answer :
Step-by-step explanation:
Sure, here are some of the most fundamental and commonly used mathematical identities across various fields:
### Algebraic Identities
1. **Square of a Binomial:**
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
2. **Product of Sum and Difference:**
\[
(a + b)(a - b) = a^2 - b^2
\]
3. **Cube of a Binomial:**
\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\]
4. **Sum and Difference of Cubes:**
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
### Trigonometric Identities
1. **Pythagorean Identities:**
\[
\sin^2\theta + \cos^2\theta = 1
\]
\[
1 + \tan^2\theta = \sec^2\theta
\]
\[
1 + \cot^2\theta = \csc^2\theta
\]
2. **Angle Sum and Difference Identities:**
\[
\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b
\]
\[
\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b
\]
\[
\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}
\]
3. **Double Angle Identities:**
\[
\sin 2\theta = 2 \sin \theta \cos \theta
\]
\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\]
\[
\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\]
4. **Half-Angle Identities:**
\[
\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}
\]
\[
\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}
\]
\[
\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}
\]
### Exponential and Logarithmic Identities
1. **Basic Exponential Identities:**
\[
a^m \cdot a^n = a^{m+n}
\]
\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\]
\[
(a^m)^n = a^{mn}
\]
2. **Basic Logarithmic Identities:**
\[
\log_b(mn) = \log_b m + \log_b n
\]
\[
\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n
\]
\[
\log_b(m^n) = n \log_b m
\]
3. **Change of Base Formula:**
\[
\log_b a = \frac{\log_k a}{\log_k b}
\]
### Complex Numbers Identities
1. **Euler's Formula:**
\[
e^{i\theta} = \cos \theta + i \sin \theta
\]
2. **De Moivre's Theorem:**
\[
(r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))
\]
### Vectors and Matrices Identities
1. **Dot Product:**
\[
\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta
\]
2. **Cross Product:**
\[
\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta \mathbf{n}
\]
3. **Matrix Multiplication:**
\[
(AB)C = A(BC)
\]
\[
A(B + C) = AB + AC
\]
\[
(A + B)C = AC + BC
\]
These identities form the foundation for solving a wide range of mathematical problems across different disciplines.