If \( A+B=45 \) , then \( (1+tan A)(1+tan B) \) is equal to

\[ Given \] \[ A + B = 45 \] \[ tan (A + B) = tan 45 \] \[ tan A + tan B/1 - tan A tan B = 1 \] \[ tan A + tan B = 1 - tan A tan B \] \[ tan A + tan B + tan A tan B = 1 \] \[ Adding 1 on both sides \] \[ 1 + tan A + tan B + tan A tan B = 2 \] \[ 1(1 + tan A) + tan B (1 + tan A) = 2 \] \[ (1 + tan A)(1 + tan B) = 2. \]

If \( A+B=45 \) , then \( (1+tan A)(1+tan... - Solved

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Question 47

Mathematics Trigonometric Equations Hard

If \( A+B=45^{\circ} \) , then \( (1+\tan A)(1+\tan B) \) is equal to

(A) 4

(B) 2

(C) 3

(D) 1

✓ Correct Answer: Option B Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\[ \text{Given} \] \[ A + B = 45^\circ \] \[ \tan (A + B) = \tan 45^\circ \] \[ \frac{\tan A + \tan B}{1 - \tan A \tan B} = 1 \] \[ \tan A + \tan B = 1 - \tan A \tan B \] \[ \tan A + \tan B + \tan A \tan B = 1 \] \[ \text{Adding } 1 \text{ on both sides} \] \[ 1 + \tan A + \tan B + \tan A \tan B = 2 \] \[ 1(1 + \tan A) + \tan B (1 + \tan A) = 2 \] \[ (1 + \tan A)(1 + \tan B) = 2. \]

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