If `T_(n)=sin^(n)theta+cos^(n)theta` then `(T_(3)-T_(5))/(T_(1))=?`

Options

A. `sintheta.costheta`
B. `sin^(2)theta.cos^(2)theta`
C. `sin^(2)theta.costheta`
D. `sintheta.cos^(2)theta`

Answer

To solve the problem, we need to evaluate the expression \((T_3 - T_5) / T_1\) where \(T_n = \sin^n \theta + \cos^n \theta\).

### Step-by-Step Solution:

1. **Define \(T_n\)**:
\[ T_n = \sin^n \theta + \cos^n \theta \]

2. **Calculate \(T_3\)**:
\[ T_3 = \sin^3 \theta + \cos^3 \theta \]

3. **Calculate \(T_5\)**:
\[ T_5 = \sin^5 \theta + \cos^5 \theta \]

4. **Calculate \(T_1\)**:
\[ T_1 = \sin \theta + \cos \theta \]

5. **Substitute into the expression**:
\[ \frac{T_3 - T_5}{T_1} = \frac{(\sin^3 \theta + \cos^3 \theta) - (\sin^5 \theta + \cos^5 \theta)}{\sin \theta + \cos \theta} \]

6. **Factor the numerator**:
- For \(T_3 - T_5\), we can factor out \(\sin^3 \theta\) and \(\cos^3 \theta\):
\[ T_3 - T_5 = \sin^3 \theta (1 - \sin^2 \theta) + \cos^3 \theta (1 - \cos^2 \theta) \]
- This simplifies to:
\[ T_3 - T_5 = \sin^3 \theta \cos^2 \theta + \cos^3 \theta \sin^2 \theta \]

7. **Rewrite the numerator**:
\[ T_3 - T_5 = \sin^2 \theta \cos^2 \theta (\sin \theta + \cos \theta) \]

8. **Substitute back into the expression**:
\[ \frac{T_3 - T_5}{T_1} = \frac{\sin^2 \theta \cos^2 \theta (\sin \theta + \cos \theta)}{\sin \theta + \cos \theta} \]

9. **Cancel out \((\sin \theta + \cos \theta)\)**:
\[ \frac{T_3 - T_5}{T_1} = \sin^2 \theta \cos^2 \theta \]

### Final Result:
\[ \frac{T_3 - T_5}{T_1} = \sin^2 \theta \cos^2 \theta \]