NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $\tan A - \tan B = x$ and $\cot B - \cot A = y$ , then $\cot (A - B)$ is equal to:

Choose the correct answer:

Explanation

Solution

Step 1: Simplify the second given equation.

We are given:

\cot B - \cot A = y

Using the identity $\cot \theta = \frac{1}{\tan \theta}$ :

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

Taking the LCM on the left side:

\frac{\tan A - \tan B}{\tan A \tan B} = y

Step 2: Substitute the value of $x$ .

From the first given equation, we know that $\tan A - \tan B = x$ . Substituting this into our simplified equation:

\frac{x}{\tan A \tan B} = y

\Rightarrow \tan A \tan B = \frac{x}{y}

Step 3: Find $\tan(A - B)$ .

Using the compound angle formula for tangent:

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

Substitute the values of $(\tan A - \tan B)$ and $(\tan A \tan B)$ :

\tan(A - B) = \frac{x}{1 + \frac{x}{y}}

\tan(A - B) = \frac{x}{\frac{y + x}{y}} = \frac{xy}{x + y}

Step 4: Find $\cot(A - B)$ .

Since $\cot(A - B) = \frac{1}{\tan(A - B)}$ :

\cot(A - B) = \frac{x + y}{xy}

Splitting the fraction:

\cot(A - B) = \frac{x}{xy} + \frac{y}{xy}

\cot(A - B) = \frac{1}{y} + \frac{1}{x}

Final Answer:

The correct option is 1: $\frac{1}{x} + \frac{1}{y}$ .