CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

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

Choose the correct answer:

Explanation

Step 1: Use the Tangent Addition Formula

We are given:

A + B = 45^\circ

Taking the tangent on both sides:

\tan(A + B) = \tan 45^\circ

We know that $\tan 45^\circ = 1$ and the expansion for $\tan(A + B)$ is:

\frac{\tan A + \tan B}{1 - \tan A \tan B} = 1

Step 2: Rearrange the Equation

Cross-multiply to remove the fraction:

\tan A + \tan B = 1 - \tan A \tan B

Now, move the term $-\tan A \tan B$ to the left side:

\tan A + \tan B + \tan A \tan B = 1

Step 3: Expand the Required Expression

Now, let's expand the expression we need to evaluate:

(1 + \tan A)(1 + \tan B) = 1 + \tan B + \tan A + \tan A \tan B

Notice that the last three terms $(\tan A + \tan B + \tan A \tan B)$ match the left side of our equation from Step 2.

Step 4: Substitute and Solve

From Step 2, we found that:

\tan A + \tan B + \tan A \tan B = 1

Substitute this value into our expanded expression:

(1 + \tan A)(1 + \tan B) = 1 + (1)

(1 + \tan A)(1 + \tan B) = 2