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If Tan α = X X + 1 and Tan α = X X + 1 , Then α + β is Equal to - CBSE (Science) Class 11 - Mathematics

ConceptTrigonometric Functions of Sum and Difference of Two Angles

Question

If $\tan\alpha = \frac{x}{x + 1}$ and $\tan\alpha = \frac{x}{x + 1}$, then $\alpha + \beta$ is equal to

• $\frac{\pi}{2}$

• $\frac{\pi}{3}$

• $\frac{\pi}{6}$

• $\frac{\pi}{4}$

Solution

It is given that $\tan\alpha = \frac{x}{x + 1}$ and

$\tan\beta = \frac{1}{2x + 1}$

$\tan\left( \alpha + \beta \right) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$

$= \frac{\frac{x}{x + 1} + \frac{1}{2x + 1}}{1 - \frac{x}{x + 1} \times \frac{1}{2x + 1}}$

$= \frac{\frac{x\left( 2x + 1 \right) + \left( x + 1 \right)}{\left( x + 1 \right)\left( 2x + 1 \right)}}{\frac{\left( x + 1 \right)\left( 2x + 1 \right) - x}{\left( x + 1 \right)\left( 2x + 1 \right)}}$

$= \frac{2 x^2 + x + x + 1}{2 x^2 + 3x + 1 - x}$

$= \frac{2 x^2 + 2x + 1}{2 x^2 + 2x + 1}$
$= 1$

$\therefore \alpha + \beta = \frac{\pi}{4} \left( \tan\frac{\pi}{4} = 1 \right)$

Hence, the correct answer is option D.

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APPEARS IN

RD Sharma Solution for Mathematics Class 11 (2019 to Current)
Chapter 7: Values of Trigonometric function at sum or difference of angles
Q: 23 | Page no. 29
Solution If Tan α = X X + 1 and Tan α = X X + 1 , Then α + β is Equal to Concept: Trigonometric Functions of Sum and Difference of Two Angles.
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