Verify :
`sec^(2)theta=1+tan^(2)theta" if "theta=(pi)/(4),`

Answer

To verify the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) for \( \theta = \frac{\pi}{4} \), we will follow these steps:

### Step 1: Calculate \( \sec \left( \frac{\pi}{4} \right) \)
The secant function is defined as the reciprocal of the cosine function. Thus, we first need to find \( \cos \left( \frac{\pi}{4} \right) \).

\[ \cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \]

Now, we can find \( \sec \left( \frac{\pi}{4} \right) \):

\[ \sec \left( \frac{\pi}{4} \right) = \frac{1}{\cos \left( \frac{\pi}{4} \right)} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 2: Calculate \( \sec^2 \left( \frac{\pi}{4} \right) \)
Now we square the value of secant:

\[ \sec^2 \left( \frac{\pi}{4} \right) = \left( \sqrt{2} \right)^2 = 2 \]

### Step 3: Calculate \( \tan \left( \frac{\pi}{4} \right) \)
Next, we find the value of the tangent function:

\[ \tan \left( \frac{\pi}{4} \right) = 1 \]

### Step 4: Calculate \( 1 + \tan^2 \left( \frac{\pi}{4} \right) \)
Now we can compute \( 1 + \tan^2 \left( \frac{\pi}{4} \right) \):

\[ \tan^2 \left( \frac{\pi}{4} \right) = 1^2 = 1 \]

Thus,

\[ 1 + \tan^2 \left( \frac{\pi}{4} \right) = 1 + 1 = 2 \]

### Step 5: Compare LHS and RHS
Now we compare the left-hand side (LHS) and right-hand side (RHS):

\[ \text{LHS} = \sec^2 \left( \frac{\pi}{4} \right) = 2 \]
\[ \text{RHS} = 1 + \tan^2 \left( \frac{\pi}{4} \right) = 2 \]

Since LHS = RHS, we have verified that:

\[ \sec^2 \theta = 1 + \tan^2 \theta \quad \text{for} \quad \theta = \frac{\pi}{4} \]

### Conclusion
Thus, the identity is verified.

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