If ` y=cos ^(-1) ((3cos x-4sin x )/( 5) ) ,then (dy)/(dx)=`

Options

A. ` 1`
B. `0`
C. `3`
D. `-4`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \cos^{-1} \left( \frac{3 \cos x - 4 \sin x}{5} \right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \cos^{-1} \left( \frac{3 \cos x - 4 \sin x}{5} \right) \]
We can express this as:
\[ y = \cos^{-1} \left( \frac{3}{5} \cos x - \frac{4}{5} \sin x \right) \]

### Step 2: Identify \( \cos \theta \) and \( \sin \theta \)
Let’s denote:
\[ \cos \theta = \frac{3}{5} \quad \text{and} \quad \sin \theta = \frac{4}{5} \]
This implies that we can form a right triangle where:
- The adjacent side (base) is 3
- The opposite side (perpendicular) is 4
- The hypotenuse is 5

### Step 3: Use the cosine of a sum identity
Using the cosine of a sum identity:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
we can rewrite \( y \) as:
\[ y = \cos^{-1} \left( \cos(\theta + x) \right) \]
Thus, we have:
\[ y = \theta + x \]

### Step 4: Differentiate
Now we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\theta + x) \]
Since \( \theta \) is a constant:
\[ \frac{dy}{dx} = 0 + 1 = 1 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 1 \]