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# If Tan θ = Sin α − Cos α Sin α + Cos α , Then Show that Sin α + Cos α = √ 2 Cos θ . - CBSE (Science) Class 11 - Mathematics

ConceptTrigonometric Functions of Sum and Difference of Two Angles

#### Question

If $\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}$ , then show that $\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta$.

#### Solution

$\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}$
Dividing numerator and denominator on the RHS by $\cos\alpha$, we get

$\tan\theta = \frac{\frac{\sin\alpha}{\cos\alpha} - 1}{\frac{\sin\alpha}{\cos\alpha} + 1}$

$\Rightarrow \tan\theta = \frac{\tan\alpha - \tan\frac{\pi}{4}}{1 + \tan\alpha \tan\frac{\pi}{4}}$

$\Rightarrow \tan\theta = \tan\left( \alpha - \frac{\pi}{4} \right)$

$\Rightarrow \theta = \alpha - \frac{\pi}{4}$

$\text{ Or }\alpha = \frac{\pi}{4} + \theta$
Now,
$\sin\alpha + \cos\alpha$
$= \sin\left( \frac{\pi}{4} + \theta \right) + \cos\left( \frac{\pi}{4} + \theta \right)$
$= \sin\frac{\pi}{4}\cos\theta + \cos\frac{\pi}{4}\sin\theta + \cos\frac{\pi}{4}\cos\theta - \sin\frac{\pi}{4}\sin\theta$
$= \frac{1}{\sqrt{2}}\cos\theta + \frac{1}{\sqrt{2}}\sin\theta + \frac{1}{\sqrt{2}}\cos\theta - \frac{1}{\sqrt{2}}\sin\theta$
$= \frac{2}{\sqrt{2}}\cos\theta$
$= \sqrt{2}\cos\theta$
$\therefore \sin\alpha + \cos\alpha = \sqrt{2}\cos\theta$

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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
Ex.7.10 | Q: 33 | Page no. 21
Solution If Tan θ = Sin α − Cos α Sin α + Cos α , Then Show that Sin α + Cos α = √ 2 Cos θ . Concept: Trigonometric Functions of Sum and Difference of Two Angles.
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