2.5 Q-7

Question Statement

We are given the equation: $y = \sqrt{\sqrt{\tan x} + \sqrt{\tan x} + \sqrt{\tan x + \ldots . \infty}}$

Prove that: $(2 y - 1) \frac{d y}{d x} = \sec^2 x$

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Background and Explanation

To solve this problem, we need to:

• Recognize the recursive nature of the equation and simplify it step by step.
• Use implicit differentiation to differentiate the equation with respect to $x$.
• Apply basic differentiation rules, including the chain rule and the derivatives of trigonometric functions.

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Solution

Step 1: Simplify the given equation

We are given a recursive equation. First, we simplify it:

$$y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \ldots}}}$$

This recursive expression suggests that we can represent the infinite nested square roots as just $y$ itself. Hence, the equation becomes:

$$y = \sqrt{\tan x + y}$$

Step 2: Square both sides

To eliminate the square root, square both sides of the equation:

$$y^2 = \tan x + y$$

Now, we have a simplified equation:

$$y^2 = \tan x + y$$

Step 3: Differentiate both sides with respect to $x$

Differentiate the equation $y^2 = \tan x + y$ with respect to $x$. We apply the chain rule on the left side and the standard derivative on the right side:

1. The derivative of $y^2$ with respect to $x$ is $2y \frac{dy}{dx}$ (using the chain rule).
2. The derivative of $\tan x$ with respect to $x$ is $\sec^2 x$.
3. The derivative of $y$ with respect to $x$ is $\frac{dy}{dx}$.

Thus, differentiating both sides gives:

$$2y \frac{dy}{dx} = \sec^2 x + \frac{dy}{dx}$$

Step 4: Solve for $\frac{dy}{dx}$

Now, rearrange the equation to solve for $\frac{dy}{dx}$:

$$2y \frac{dy}{dx} - \frac{dy}{dx} = \sec^2 x$$

Factor out $\frac{dy}{dx}$ on the left-hand side:

$$(2y - 1) \frac{dy}{dx} = \sec^2 x$$

Step 5: Conclusion

We have shown that:

$$(2y - 1) \frac{dy}{dx} = \sec^2 x$$

Thus, we have successfully proven the required result.

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Key Formulas or Methods Used

• Chain Rule: For differentiating composite functions.

$$\frac{d}{dx} (y^2) = 2y \frac{dy}{dx}$$

• Derivative of $\tan x$:

$$\frac{d}{dx} (\tan x) = \sec^2 x$$

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Summary of Steps

1. Simplify the recursive equation by representing the infinite nested square roots as $y$.
2. Square both sides of the equation to eliminate the square root.
3. Differentiate both sides of the equation with respect to $x$.
4. Rearrange and solve for $\frac{dy}{dx}$.
5. Conclude that $(2y - 1) \frac{dy}{dx} = \sec^2 x$.

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