Question

If cosA + sinA = √2 cosA, Show that cosA – sinA = √2 sinA​

Answer 1

Step-by-step explanation:

cos A +sin A = √2cos A

squaring on both sides we get

(cos A + sin A)² = (√2cos A)²

it is in the form of (a+b)^2

formula for (a+b)^2=a^2+ab+b^2

by comparing here we have a=cos A and b=sin A

cos²A + sin²A + 2sinAcosA = 2cos²A

we know that

sin^2A+cos^2 A=1

from this we can write

sin^2 A=1-cos^2 A

and cos^2 A=1-sin^2 A

1 - sin²A + 1 - cos²A + 2 sinAcosA = 2 cos²A

adding like terms and sin^2 A+cos^2 A -2 sinA cosA -2 cos^2 A on both sides

2 - 2 cos²A = cos²A + sin²A - 2 sinAcosA  

taking 2 as common in left side of the equation

2(1 - cos²A)= (cos A - sin A)²

cos A - sin A = √[2 sin²A]

cos A-sin A = √2 sin A

hence it is proved

Answer 2

Answer:

sin A+cosA=√2cosA

Step-by-step explanation:

cosA-sinA=√2sinA

to prove cosA+sinA

cosA=√2sinA+sinA

cosA=sinA(√2+1)

sinA=cosA/√2-1

by rationalisation cosA/√2+1*√2-1/√2-1

sinA=cos√2-cosA/1

sinA+cosA=√2cosA