Question

The value of \( \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} \) will be

• A. -1
• B. 0
• C. 1
• D. 2

Hint:

Memorizing key trigonometric identities like the double angle formulas can provide a very quick shortcut for solving such problems. Recognizing the pattern \( \frac{2 \tan \theta}{1 + \tan^2 \theta} \) immediately tells you the answer is \( \sin(2\theta) \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires the evaluation of a trigonometric expression. We need to know the value of \( \tan 45^\circ \) and substitute it into the given expression.
Step 2: Key Formula or Approach:
We will use the known value of \( \tan 45^\circ = 1 \).
Alternatively, we can use the trigonometric identity for sine of a double angle: \( \sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta} \).
Step 3: Detailed Explanation:
Method 1: Direct Substitution
We know that the value of \( \tan 45^\circ \) is 1.
Substitute this value into the expression:
\[ \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} = \frac{2 \times (1)}{1 + (1)^2} \] \[ = \frac{2}{1 + 1} \] \[ = \frac{2}{2} \] \[ = 1 \] Method 2: Using Double Angle Identity
The given expression is in the form of the identity for \( \sin(2\theta) \).
\[ \sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta} \] In this problem, \( \theta = 45^\circ \).
So, the expression is equal to \( \sin(2 \times 45^\circ) \).
\[ \sin(90^\circ) \] We know that \( \sin(90^\circ) = 1 \).
Step 4: Final Answer:
The value of the expression is 1.