If ` x = sin theta, y = tan theta ` then find ` (dy)/(dx) `

Answer

To find \(\frac{dy}{dx}\) given \(x = \sin \theta\) and \(y = \tan \theta\), we can use the chain rule. Here’s the step-by-step solution:

### Step 1: Express \(\frac{dy}{dx}\) using the chain rule
We can express \(\frac{dy}{dx}\) in terms of \(\theta\):
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

### Step 2: Differentiate \(x\) with respect to \(\theta\)
Given \(x = \sin \theta\), we differentiate:
\[ \frac{dx}{d\theta} = \cos \theta \]

### Step 3: Differentiate \(y\) with respect to \(\theta\)
Given \(y = \tan \theta\), we differentiate:
\[ \frac{dy}{d\theta} = \sec^2 \theta \]

### Step 4: Substitute the derivatives into the chain rule expression
Now we substitute \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\) into the expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\sec^2 \theta}{\cos \theta} \]

### Step 5: Simplify the expression
Since \(\sec \theta = \frac{1}{\cos \theta}\), we can rewrite \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{\sec^2 \theta}{\cos \theta} = \sec^2 \theta \cdot \frac{1}{\cos \theta} = \sec^3 \theta \]

### Final Answer
Thus, the final result is:
\[ \frac{dy}{dx} = \sec^3 \theta \]

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