If the rate of change of sine of an angle `theta` is k, then the rate of change of its tangent is

Options

A. `k^(2)`
B. `(1)/(k^(2))`
C. k
D. `(1)/(k)`

Answer

To solve the problem, we need to find the rate of change of the tangent of an angle \(\theta\) given that the rate of change of the sine of the angle is \(k\).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that the rate of change of sine of an angle \(\theta\) is \(k\). Mathematically, this can be expressed as:
\[ \frac{d}{d\theta}(\sin \theta) = k \]

2. **Using the Derivative of Sine**:
We know from calculus that the derivative of \(\sin \theta\) with respect to \(\theta\) is:
\[ \frac{d}{d\theta}(\sin \theta) = \cos \theta \]
Therefore, we can equate:
\[ \cos \theta = k \]

3. **Finding the Rate of Change of Tangent**:
We need to find the rate of change of \(\tan \theta\). The derivative of \(\tan \theta\) with respect to \(\theta\) is:
\[ \frac{d}{d\theta}(\tan \theta) = \sec^2 \theta \]

4. **Relating Secant to Cosine**:
We can express \(\sec^2 \theta\) in terms of \(\cos \theta\):
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]

5. **Substituting the Value of Cosine**:
Since we found that \(\cos \theta = k\), we can substitute this into the equation:
\[ \sec^2 \theta = \frac{1}{k^2} \]

6. **Final Result**:
Therefore, the rate of change of the tangent of the angle \(\theta\) is:
\[ \frac{d}{d\theta}(\tan \theta) = \frac{1}{k^2} \]

### Conclusion:
The rate of change of the tangent of the angle \(\theta\) is \(\frac{1}{k^2}\).

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