If `(costheta)/(1+sintheta)+(coshtheta)/(1-sintheta)=4andtheta` is acute , then what is the value (in degrees) of `theta` ?

Options

A. 30
B. 45
C. 60
D. 90

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
We start by finding a common denominator for the two fractions on the left side of the equation. 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, we simplify the numerator:

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

So, we rewrite the equation as:

\[ \frac{2 \cos \theta}{(1 + \sin \theta)(1 - \sin \theta)} = 4. \]

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

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

Thus, the equation becomes:

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

### Step 4: Cross-multiply
Cross-multiplying gives us:

\[ 2 \cos \theta = 4 \cos^2 \theta. \]

### Step 5: Rearranging the equation
Rearranging the equation leads to:

\[ 4 \cos^2 \theta - 2 \cos \theta = 0. \]

### Step 6: Factor the equation
Factoring out \(2 \cos \theta\):

\[ 2 \cos \theta (2 \cos \theta - 1) = 0. \]

This gives us two possible solutions:

1. \(2 \cos \theta = 0\) which leads to \(\cos \theta = 0\) (not valid since \(\theta\) is acute).
2. \(2 \cos \theta - 1 = 0\) which leads to \(\cos \theta = \frac{1}{2}\).

### Step 7: Find the angle
The angle whose cosine is \(\frac{1}{2}\) in the acute range is:

\[ \theta = 60^\circ. \]

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

\[ \theta = 60^\circ. \]

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