If `x=sectheta-costheta` and `y=sec^n theta- cos^n theta` then show that `(x^2+4)((dy)/(dx))^2=n^2(y^2+4)`

Options

A. 8
B. 16
C. 64
D. 49

Explanation

To solve the problem, we need to show that:

\[ (x^2 + 4) \left( \frac{dy}{dx} \right)^2 = n^2 (y^2 + 4) \]

where \( x = \sec \theta - \cos \theta \) and \( y = \sec^n \theta - \cos^n \theta \).

### Step 1: Find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\)

First, we differentiate \(y\) and \(x\) with respect to \(\theta\).

1. **Differentiate \(y\)**:
\[ y = \sec^n \theta - \cos^n \theta \]
Using the chain rule:
\[ \frac{dy}{d\theta} = n \sec^{n} \theta \tan \theta - n \cos^{n-1} \theta \cdot (-\sin \theta) \]
Simplifying this gives:
\[ \frac{dy}{d\theta} = n \sec^{n} \theta \tan \theta + n \cos^{n-1} \theta \sin \theta \]

2. **Differentiate \(x\)**:
\[ x = \sec \theta - \cos \theta \]
Again using the chain rule:
\[ \frac{dx}{d\theta} = \sec \theta \tan \theta + \sin \theta \]

### Step 2: Find \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]

Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{n \sec^{n} \theta \tan \theta + n \cos^{n-1} \theta \sin \theta}{\sec \theta \tan \theta + \sin \theta} \]

### Step 3: Simplify \(\frac{dy}{dx}\)

Notice that:
\[ \frac{dy}{dx} = n \cdot \frac{\sec^n \theta - \cos^n \theta}{\sec \theta - \cos \theta} = n \cdot \frac{y}{x} \]

### Step 4: Square \(\frac{dy}{dx}\)

Now we take the square:
\[ \left( \frac{dy}{dx} \right)^2 = n^2 \cdot \left( \frac{y}{x} \right)^2 \]

### Step 5: Substitute into the original equation

We substitute this back into the equation we need to prove:
\[ (x^2 + 4) \left( \frac{dy}{dx} \right)^2 = (x^2 + 4) n^2 \cdot \frac{y^2}{x^2} \]

### Step 6: Use the component or dividendo

Using the component or dividendo:
\[ \frac{y^2 + 4}{x^2 + 4} = \frac{y^2}{x^2} \]

Thus, we can write:
\[ (x^2 + 4) \left( \frac{dy}{dx} \right)^2 = n^2 (y^2 + 4) \]

### Conclusion

We have shown that:
\[ (x^2 + 4) \left( \frac{dy}{dx} \right)^2 = n^2 (y^2 + 4) \]

This completes the proof.