Find the value of `theta`, if `(costheta)/(1-sintheta)+(costheta)/(1+sintheta)=4,thetale90^(@)`.

Answer

To solve the equation

\[ \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4 \]

we will follow these steps:

### Step 1: Combine the fractions
To combine the two fractions on the left-hand side, we need a common denominator. The common denominator is \((1 - \sin \theta)(1 + \sin \theta)\).

\[ \frac{\cos \theta (1 + \sin \theta) + \cos \theta (1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = 4 \]

### Step 2: Simplify the numerator
Now, simplifying the numerator:

\[ \cos \theta (1 + \sin \theta) + \cos \theta (1 - \sin \theta) = \cos \theta + \cos \theta \sin \theta + \cos \theta - \cos \theta \sin \theta = 2 \cos \theta \]

### Step 3: Simplify the denominator
The denominator can be simplified using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):

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

### Step 4: Rewrite the equation
Now we can rewrite the equation:

\[ \frac{2 \cos \theta}{\cos^2 \theta} = 4 \]

### Step 5: Simplify the equation
This simplifies to:

\[ \frac{2}{\cos \theta} = 4 \]

### Step 6: Solve for \(\cos \theta\)
Cross-multiplying gives:

\[ 2 = 4 \cos \theta \]

Dividing both sides by 4:

\[ \cos \theta = \frac{1}{2} \]

### Step 7: Find \(\theta\)
The value of \(\theta\) for which \(\cos \theta = \frac{1}{2}\) is:

\[ \theta = 60^\circ \]

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

\[ \theta = 60^\circ \]

---