Question

if sinФ +cosФ= √2 cosФ, (Ф≠90°) then value of tan is ?

Answer 1

Given :

$\bf \sin \theta + \cos \theta = \sqrt{2} \cos\theta$

To Find :

$\bf Value \: of \: \tan\theta$

Solution :

Method 1

Both side divided by Cos Ø

$\implies \bf\dfrac{ \sin \theta + \cos \theta}{ \cos \theta} = \dfrac{ \sqrt{2} \cos \theta }{ \cos \theta} \\ \\\bf \implies\dfrac{ \sin \theta}{ \cos \theta} + \dfrac{ \cancel{ \cos \theta}}{\cancel{ \cos \theta}} = \dfrac{ \sqrt{2}\cancel{ \cos \theta }}{ \cancel{\cos \theta} } \\ \\ \bf \implies \tan \theta + 1 = \sqrt{2} \\ \\ \large \implies\boxed{ \bf \tan\theta= \sqrt{2} - 1}$

Method 2

Squaring both sides

$\bf \implies{(\sin\theta + \cos \theta)}^{2} = {(\sqrt{2}\cos\theta)}^{2}\\ \\ \implies \bf {\sin\theta}^{2}+ {\cos\theta}^{2}+2\sin\theta\cos\theta = {2 \cos\theta}^{2}\\ \\ \bf \implies 1 + 2 \sin\theta\cos\theta = {2 \cos\theta}^{2}\\ \\ \bf \implies \sin2\theta= \cos2\theta \\ \\ \bf \implies \sin(2\theta)= \cos(90 - 2\theta )\\ \\ \bf \implies 2 \theta = 90 - 2\theta \\ \\\bf \implies 4 \theta = 90 \\ \\ \bf \implies\theta = 22.5 \\ \\\large \implies\boxed{ \bf \tan\theta= \sqrt{2} - 1}$

Answer 2

Answer:

Given:

sinθ + cosθ = √2cosθ

to find:

value of tanθ

solution:

squaring both sides-->

(sinθ + cosθ)^2 = (√2cosθ)^2

sinθ^2 + cosθ ^2 +2sinθcosθ= 2cosθ^2

=> 1+ 2sinθcosθ = 2cosθ^2

=> sin2θ = cos2θ

=>2θ = (90-2θ )

=> 4θ = 90

=>θ = 22.5°

=> tan θ = √2-1