$(\frac{cos~ heta+i~sin~ heta}{cos~ heta-i~sin~ heta})^{4}=$

Q: $(\frac{cos~\theta+i~sin~\theta}{cos~\theta-i~sin~\theta})^{4}=$

$(\frac{cos~\theta+i~sin~\theta}{cos~\theta-i~sin~\theta})^{4}=$

Question

$(\frac{cos~	heta+i~sin~	heta}{cos~	heta-i~sin~	heta})^{4}=$

Accepted Answer

To solve the given expression $(\frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta})^{4}$, we need to simplify it and verify why Option 4 is the correct answer.

### Explanation:

1. **Simplification of the Expression:**

The expression $\frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta}$ can be simplified using the properties of complex numbers. This expression is a form of a complex fraction, which can be simplified by multiplying the numerator and the denominator by the conjugate of the denominator:

$$
   \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \times \frac{\cos \theta + i \sin \theta}{\cos \theta + i \sin \theta} = \frac{(\cos \theta + i \sin \theta)^2}{\cos^2 \theta + \sin^2 \theta}
   $$

Since $\cos^2 \theta + \sin^2 \theta = 1$, the expression simplifies to:

$$
   (\cos \theta + i \sin \theta)^2
   $$

2. **Using De Moivre's Theorem:**

According to De Moivre's Theorem, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$. Therefore, for $n = 2$:

$$
   (\cos \theta + i \sin \theta)^2 = \cos(2\theta) + i \sin(2\theta)
   $$

3. **Raising to the Fourth Power:**

Now, we need to raise the result to the fourth power:

$$
   (\cos(2\theta) + i \sin(2\theta))^4
   $$

Applying De Moivre's Theorem again:

$$
   = \cos(4 \times 2\theta) + i \sin(4 \times 2\theta) = \cos(8\theta) + i \sin(8\theta)
   $$

This matches Option 4: $ \cos 8\theta + i \sin 8\theta $.

### Why Other Options are Incorrect:

- **Option 1: 1**

This option would imply that the expression simplifies to a real number 1, which is incorrect because the expression involves complex numbers and results in a complex number with both cosine and sine components.

- **Option 2: 0**

This option suggests that the expression evaluates to zero, which is incorrect. The expression involves trigonometric functions and complex numbers, and it does not simplify to zero.

- **Option 3: $ \cos 4\theta - i \sin 4\theta $**

This option is incorrect because it suggests a different angle and incorrect sign for the imaginary part. The correct simplification involves $8\theta$ and a positive imaginary component.

Thus, the correct answer is Option 4: $ \cos 8\theta + i \sin 8\theta $.