If sinx + cosx = √2cosx, what will be the value of sinx - cosx ?

If sinx + cosx = √2cosx, what will be the value of sinx - cosx ? Correct Answer <u>+</u> √2sinx

Given:

sinx + cosx = √2cosx

Concept Used:

sin²x + cos²x = 1

Calculation:

sinx + cosx = √2cosx

Squaring the equation on both side, we get

⇒ sin²x + cos²x +2sinxcosx = 2cos²x

⇒ 1 + 2sinxcosx = 2cos²x      ----(i)

Let sinx - cosx = k      ----(ii)

Squaring the equation on both side, we get

⇒ 1 - 2sinxcosx = k²

Adding (i) and (ii), we get

2cos²x + k² = 2

⇒ k² = 2(1 - cos²x)

⇒ k² = 2sin²x

⇒ k = +√2sinx

∴ The value of sinx - cosx = +√2sinx