Solve : If `(1)/(cos theta)+(1)/(cos 3theta)=(1)/(cos 5theta)`

Explanation

To solve the equation

\[ \frac{1}{\cos \theta} + \frac{1}{\cos 3\theta} = \frac{1}{\cos 5\theta}, \]

we will follow these steps:

### Step 1: Take LCM and Cross Multiply
We start by taking the least common multiple (LCM) of the fractions on the left-hand side:

\[ \frac{\cos 3\theta + \cos \theta}{\cos \theta \cos 3\theta} = \frac{1}{\cos 5\theta}. \]

Cross multiplying gives us:

\[ (\cos 3\theta + \cos \theta) \cos 5\theta = \cos \theta \cos 3\theta. \]

### Step 2: Expand the Equation
Now, we can expand the equation:

\[ \cos 3\theta \cos 5\theta + \cos \theta \cos 5\theta = \cos \theta \cos 3\theta. \]

### Step 3: Rearranging the Terms
Rearranging the equation gives us:

\[ \cos 3\theta \cos 5\theta + \cos \theta \cos 5\theta - \cos \theta \cos 3\theta = 0. \]

### Step 4: Factor Out Common Terms
We can factor out \(\cos \theta\):

\[ \cos 5\theta (\cos 3\theta + \cos \theta) - \cos \theta \cos 3\theta = 0. \]

### Step 5: Apply the Cosine Addition Formula
Using the identity \(2 \cos A \cos B = \cos(A + B) + \cos(A - B)\), we can rewrite the terms:

\[ 2 \cos 3\theta \cos 5\theta + 2 \cos \theta \cos 5\theta - 2 \cos \theta \cos 3\theta = 0. \]

### Step 6: Simplifying the Equation
This simplifies to:

\[ \cos(8\theta) + \cos(2\theta) = 0. \]

### Step 7: Using the Cosine Sum Formula
Using the cosine sum formula:

\[ \cos A + \cos B = 0 \Rightarrow 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) = 0. \]

Setting \(A = 8\theta\) and \(B = 2\theta\):

\[ 2 \cos(5\theta) \cos(3\theta) = 0. \]

### Step 8: Setting Each Factor to Zero
This gives us two cases:

1. \(\cos(5\theta) = 0\)
2. \(\cos(3\theta) = 0\)

### Step 9: Solving Each Case
For \(\cos(5\theta) = 0\):

\[ 5\theta = \frac{(2n + 1)\pi}{2} \Rightarrow \theta = \frac{(2n + 1)\pi}{10}, \quad n \in \mathbb{Z}. \]

For \(\cos(3\theta) = 0\):

\[ 3\theta = \frac{(2m + 1)\pi}{2} \Rightarrow \theta = \frac{(2m + 1)\pi}{6}, \quad m \in \mathbb{Z}. \]

### Final Solution
Thus, the solutions for \(\theta\) are:

\[ \theta = \frac{(2n + 1)\pi}{10} \quad \text{and} \quad \theta = \frac{(2m + 1)\pi}{6}. \]