Question

If cos A+sin A=√2 cosA, show that cos A-sin A=√2sinA

Answer 1

$\cos(a) + \sin(a) = \sqrt{2} \cos(a) \\ {( \cos(a) + \sin(a) ) }^{2} = 2 { (\cos(a) )}^{2} \\ { \cos(a) }^{2} + { \sin(a) }^{2} + 2 \cos(a) \sin(a) = 2 { \cos(a) }^{2} \\ 2 \cos(a) \sin(a) = 2 { \cos(a) }^{2} - 1 \\ \cos(a) - \sin(a) = \sqrt{ {( \cos(a) + \sin(a)) }^{2} - 4 \cos(a) \sin(a) } \\ = \sqrt{2 { \cos(a) }^{2} - 2(2 { \cos(a) }^{2} - 1) } \\ = \sqrt{2 { \cos(a) }^{2} - 4 { \cos(a) }^{2} + 2} \\ = \sqrt{2 - 2 { \cos(a) }^{2} } \\ = \sqrt{ 2(1 - { \cos(a) }^{2} )} \\ = \sqrt{2 { \sin(a) }^{2} } \\ = \sqrt{2} \sin(a)$

Answer 2

Given: cosA+sinA=√2cosA
To prove: cosA-sinA=√2sinA
Proof:
(cosA+sinA)²= sin²A+cos²A+2sinA.cosA
(√2cosA)² = 1+2sinA.cosA
2sinA.cosA= 2cos²A-1

(cosA-sinA)²= sin²A+cos²A-2sinA.cosA
(cosA-sinA)²= 1-2sinA.cosA
(cosA-sinA)²= 1-(2cos²A-1)
(cosA-sinA)²= 1+1-2cos²A
(cosA-sinA)²= 2-2cos²A
(cosA-sinA)²= 2(1-cos²A)
(cosA-sinA)²= 2(sin²A)
(cosA-sinA) = √2sinA.
Thus proved.