If `3f(cosx)+2f(sinx)=5x`, then `f^(')(cosx)` is equal to (where `f^(')` denotes derivative with respect to`x`) (A) `−1/(cosx)` (B) `1/(cosx)` (C) `-1/(sinx)` (D) `1/(sinx)`

Options

A. `-5/(cosx)`
B. `5/(cosx)`
C. `-5/(sinx)`
D. `5/(sinx)`

Explanation

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ 3f(\cos x) + 2f(\sin x) = 5x \tag{1} \]

2. **Substituting \( x \) with \( \frac{\pi}{2} - x \)**:
We replace \( x \) in equation (1) with \( \frac{\pi}{2} - x \):
\[ 3f\left(\cos\left(\frac{\pi}{2} - x\right)\right) + 2f\left(\sin\left(\frac{\pi}{2} - x\right)\right) = 5\left(\frac{\pi}{2} - x\right) \]
Using the trigonometric identities \( \cos\left(\frac{\pi}{2} - x\right) = \sin x \) and \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \), we rewrite the equation:
\[ 3f(\sin x) + 2f(\cos x) = 5\left(\frac{\pi}{2} - x\right) \tag{2} \]

3. **Multiply Equation (1) by 3 and Equation (2) by 2**:
- From (1):
\[ 9f(\cos x) + 6f(\sin x) = 15x \]
- From (2):
\[ 6f(\sin x) + 4f(\cos x) = 5\pi - 10x \]

4. **Subtract the modified equations**:
Now, we subtract the second modified equation from the first:
\[ (9f(\cos x) + 6f(\sin x)) - (6f(\sin x) + 4f(\cos x)) = 15x - (5\pi - 10x) \]
This simplifies to:
\[ 5f(\cos x) = 15x - 5\pi + 10x \]
\[ 5f(\cos x) = 25x - 5\pi \]
Dividing through by 5:
\[ f(\cos x) = 5x - \pi \tag{3} \]

5. **Express \( f(t) \)**:
Let \( t = \cos x \). Then, from equation (3):
\[ f(t) = 5\cos^{-1}(t) - \pi \]

6. **Differentiate \( f(t) \)**:
Now we differentiate \( f(t) \) with respect to \( t \):
\[ f'(t) = \frac{d}{dt}(5\cos^{-1}(t) - \pi) \]
The derivative of \( \cos^{-1}(t) \) is \( -\frac{1}{\sqrt{1 - t^2}} \):
\[ f'(t) = 5 \cdot \left(-\frac{1}{\sqrt{1 - t^2}}\right) = -\frac{5}{\sqrt{1 - t^2}} \]

7. **Substituting back \( t = \cos x \)**:
Now we substitute back \( t = \cos x \):
\[ f'(\cos x) = -\frac{5}{\sqrt{1 - \cos^2 x}} = -\frac{5}{\sin x} \]

8. **Final Result**:
Therefore, we have:
\[ f'(\cos x) = -\frac{5}{\sin x} \]

Since the question asks for \( f'(\cos x) \) and we have derived that it equals \( -\frac{5}{\sin x} \), we can conclude that the correct answer is:

**Answer**: (C) \(-\frac{1}{\sin x}\)