Question

if cos ∅ - sin ∅ = √2 sin ∅, prove that cos ∅ + sin ∅
= √2 cos∅ prove

Answer 1

Hola there,

Let ∅ be 'A'

Given=> cosA - sinA = √2 sinA

To prove => cosA + sinA = √2 cosA

Proof =>

(cosA - sinA)² = (√2 sinA)²

=> cos²A + sin²A - 2sinAcosA = 2sin²A

=> 2sin²A - sin²A - cos²A = 2sinAcosA

=> cos²A = sin²A + 2sinAcosA

Adding cos²A both sides, we get

=> 2cos²A = sin²A + 2sinAcosA + cos²A

=> 2cos²A = (cosA + sinA)²

=> cosA + sinA = √2 cosA

Hence Proved

Hope this helps...:)

Answer 2

Hey,

$cos \alpha - sin \alpha = \sqrt{2} sin \alpha$

$= > cos \alpha = ( \sqrt{2} + 1)sin \alpha$

$= > sin \alpha = \frac{1}{( \sqrt{2} + 1) } cos \alpha$


Rationalize the denominator to get :
$\sin( \alpha ) = ( \sqrt{2} - 1) \cos( \alpha )$

Bring cos to other side to get :

$\sin( \alpha ) + \cos( \alpha ) = \sqrt{2} \cos( \alpha )$