Find the equations of the tangent and normal lines to the following curves :
`y=sin^(2)x" at "x=(pi)/(2).`

Answer

To find the equations of the tangent and normal lines to the curve \( y = \sin^2 x \) at the point where \( x = \frac{\pi}{2} \), we will follow these steps:

### Step 1: Find the coordinates of the point on the curve
We need to evaluate the function at \( x = \frac{\pi}{2} \):
\[ y = \sin^2\left(\frac{\pi}{2}\right) \]
Since \( \sin\left(\frac{\pi}{2}\right) = 1 \):
\[ y = 1^2 = 1 \]
Thus, the coordinates of the point are:
\[ \left(\frac{\pi}{2}, 1\right) \]

### Step 2: Find the derivative of the function
To find the slope of the tangent line, we differentiate \( y = \sin^2 x \) using the chain rule:
\[ \frac{dy}{dx} = 2\sin x \cdot \cos x \]
This can also be written as:
\[ \frac{dy}{dx} = \sin(2x) \]

### Step 3: Evaluate the derivative at \( x = \frac{\pi}{2} \)
Now we substitute \( x = \frac{\pi}{2} \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \sin\left(2 \cdot \frac{\pi}{2}\right) = \sin(\pi) = 0 \]
Thus, the slope of the tangent line at this point is \( 0 \).

### Step 4: Write the equation of the tangent line
The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point of tangency and \( m \) is the slope. Here, \( (x_1, y_1) = \left(\frac{\pi}{2}, 1\right) \) and \( m = 0 \):
\[ y - 1 = 0 \cdot \left(x - \frac{\pi}{2}\right) \]
This simplifies to:
\[ y = 1 \]
So, the equation of the tangent line is:
\[ y = 1 \]

### Step 5: Find the slope of the normal line
The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line is \( 0 \), the slope of the normal line is undefined, which means it is a vertical line.

### Step 6: Write the equation of the normal line
Since the normal line is vertical and passes through the point \( \left(\frac{\pi}{2}, 1\right) \), its equation is:
\[ x = \frac{\pi}{2} \]

### Final Answer
The equations of the tangent and normal lines are:
- Tangent line: \( y = 1 \)
- Normal line: \( x = \frac{\pi}{2} \)