Question

SinA+ cos A=√2sinA then,sinA-cosA=

Answer 1

Given :

• Sin A + Cos A = √2 sin A

To find :

• Sin A - cos A

Solution :

Sin A + Cos A = √2 sin A

• Squaring both the sides :

(sin A + cos A)² = (√2 sin A)²

• Solving by using Identity :
• ( x + y)² = x² + y² + 2xy

Sin² A + Cos² A + 2Sin A Cos A = 2 sin²A ____(1)

Using Identity

Sin² A + cos ² A = 1

Sin² A = 1 - cos² A ______(2)

Using equation 2 jinin n equation 1 :

1 - cos² A + cos² A + 2 sin A cos A = 2 ( 1 - cos²A)

1 + 2 sin A cos A = 2 - 2 cos² A

2 sin A cos A =  2 - 1 - 2 cos² A

2 sin A cos A = 1  - 2cos² A

2cos² A = 1 - 2 sin A cos A

2Cos² A =  sin² A + cos² A - 2 sin A cos A

Now , Using identity

( x - y)² = x² + y² - 2xy

2Cos² A = ( sin A - cos A)²

Sin A - cos A = √(2 cos² A)

Sin A - cos A = ± √2 cos A