Find the value of `(1)/((1+tan^(2)theta))+(1)/((1+cot^(2)theta))`

Options

A. a) 2
B. b) `(1)/(2)`
C. c) `(1)/(4)`
D. d) 1

Answer

To solve the expression \(\frac{1}{1+\tan^2\theta} + \frac{1}{1+\cot^2\theta}\), we can follow these steps:

### Step 1: Use Trigonometric Identities
We know from trigonometric identities that:
\[ 1 + \tan^2\theta = \sec^2\theta \]
and
\[ 1 + \cot^2\theta = \csc^2\theta \]

### Step 2: Substitute the Identities
Substituting these identities into the expression, we get:
\[ \frac{1}{1+\tan^2\theta} = \frac{1}{\sec^2\theta} = \cos^2\theta \]
and
\[ \frac{1}{1+\cot^2\theta} = \frac{1}{\csc^2\theta} = \sin^2\theta \]

### Step 3: Combine the Results
Now, we can rewrite the original expression:
\[ \frac{1}{1+\tan^2\theta} + \frac{1}{1+\cot^2\theta} = \cos^2\theta + \sin^2\theta \]

### Step 4: Apply the Pythagorean Identity
Using the Pythagorean identity:
\[ \sin^2\theta + \cos^2\theta = 1 \]

### Final Result
Thus, the value of the expression is:
\[ \frac{1}{1+\tan^2\theta} + \frac{1}{1+\cot^2\theta} = 1 \]

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