The angle made by any tangent to the curve `x=a(t+sintcost) , y=(1+sint)^2` with `x`-axis is:

Options

A. `1/4(pi+2t)`
B. `(1-sin t)/(cost)`
C. `1/4(2t-pi)`
D. `(1+sint)/(cos 2t)`

Answer

To find the angle made by the tangent to the curve defined by the equations \( x = a(t + \sin t \cos t) \) and \( y = a(1 + \sin t)^2 \) with the x-axis, we can follow these steps:

### Step 1: Differentiate the Parametric Equations
We need to find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).

1. **Differentiate \( x \)**:
\[ x = a(t + \sin t \cos t) \]
Using the product rule on \( \sin t \cos t \):
\[ \frac{dx}{dt} = a\left(1 + \frac{d}{dt}(\sin t \cos t)\right) = a\left(1 + \cos^2 t - \sin^2 t\right) = a(1 + \cos 2t) \]

2. **Differentiate \( y \)**:
\[ y = a(1 + \sin t)^2 \]
Using the chain rule:
\[ \frac{dy}{dt} = a \cdot 2(1 + \sin t) \cdot \cos t = 2a(1 + \sin t)\cos t \]

### Step 2: Find the Slope of the Tangent
The slope of the tangent line at any point on the curve is given by:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{2a(1 + \sin t)\cos t}{a(1 + \cos 2t)} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{2(1 + \sin t)\cos t}{1 + \cos 2t} \]

### Step 3: Relate the Slope to the Angle
The angle \( \theta \) that the tangent makes with the x-axis can be found using:
\[ \tan \theta = \frac{dy}{dx} \]
Thus:
\[ \tan \theta = \frac{2(1 + \sin t)\cos t}{1 + \cos 2t} \]

### Step 4: Simplify the Expression
Using the double angle identity \( \cos 2t = 1 - 2\sin^2 t \):
\[ 1 + \cos 2t = 2\cos^2 t \]
So we can rewrite:
\[ \tan \theta = \frac{2(1 + \sin t)\cos t}{2\cos^2 t} = \frac{(1 + \sin t)}{\cos t} \]

### Step 5: Find the Angle
To find \( \theta \):
\[ \tan \theta = \frac{1 + \sin t}{\cos t} \]
This can be expressed as:
\[ \tan \theta = \tan\left(\frac{\pi}{4} + \frac{t}{2}\right) \]
Thus, we can conclude:
\[ \theta = \frac{\pi}{4} + \frac{t}{2} \]

### Final Answer
The angle made by the tangent to the curve with the x-axis is:
\[ \theta = \frac{\pi}{4} + \frac{t}{2} \]