If `(sin^(2)theta)/(1+cos^(2)theta)+(sin^(2)theta)/(1-cos^(2)theta)=(8)/(5)andtheta` is acute , then what is the value (in degrees ) of `theta` ?

Options

A. 0
B. 30
C. 45
D. 60

Answer

To solve the equation

\[ \frac{\sin^2 \theta}{1 + \cos^2 \theta} + \frac{\sin^2 \theta}{1 - \cos^2 \theta} = \frac{8}{5} \]

we will follow these steps:

### Step 1: Simplify the equation
We know that \(1 - \cos^2 \theta = \sin^2 \theta\). Therefore, we can rewrite the second term:

\[ \frac{\sin^2 \theta}{1 + \cos^2 \theta} + \frac{\sin^2 \theta}{\sin^2 \theta} = \frac{8}{5} \]

This simplifies to:

\[ \frac{\sin^2 \theta}{1 + \cos^2 \theta} + 1 = \frac{8}{5} \]

### Step 2: Isolate the first term
Subtract 1 from both sides:

\[ \frac{\sin^2 \theta}{1 + \cos^2 \theta} = \frac{8}{5} - 1 \]

Calculating the right side:

\[ \frac{8}{5} - \frac{5}{5} = \frac{3}{5} \]

So we have:

\[ \frac{\sin^2 \theta}{1 + \cos^2 \theta} = \frac{3}{5} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:

\[ 5 \sin^2 \theta = 3 (1 + \cos^2 \theta) \]

### Step 4: Expand and rearrange
Expanding the right side:

\[ 5 \sin^2 \theta = 3 + 3 \cos^2 \theta \]

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can express \(\cos^2 \theta\) as \(1 - \sin^2 \theta\):

\[ 5 \sin^2 \theta = 3 + 3(1 - \sin^2 \theta) \]

This simplifies to:

\[ 5 \sin^2 \theta = 3 + 3 - 3 \sin^2 \theta \]

Combining like terms:

\[ 5 \sin^2 \theta + 3 \sin^2 \theta = 6 \]

### Step 5: Combine and solve for \(\sin^2 \theta\)
This gives us:

\[ 8 \sin^2 \theta = 6 \]

Dividing both sides by 8:

\[ \sin^2 \theta = \frac{6}{8} = \frac{3}{4} \]

### Step 6: Find \(\sin \theta\)
Taking the square root:

\[ \sin \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 7: Determine \(\theta\)
The angle \(\theta\) for which \(\sin \theta = \frac{\sqrt{3}}{2}\) is:

\[ \theta = 60^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:

\[ \theta = 60^\circ \]

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