If sin x + cos x = √2 sin x, then the value of sin x -

Solution

sinx + cosx = √2sinx Squaring the equation on both sides, we get. ⇒ sin²x + cos² x +2sinxcosx = 2sin² x ⇒ 1 + 2sinx.cosx = 2sin² x     ---- (i) Let sinx - cosx = k      ---- (ii) Squaring the equation on both sides, we get. ⇒ 1 - 2sinxcosx = k² Adding (i) and (ii), we get 2sin² x + k² = 2     k² = 2(1-sin²x)   k² = 2cos² x   k = ± √ (2cos² x) The value of (sinx – cosx) = ± 2√cosx Alternate- Concept- if (sinx – cosx) =k then (sinx – cosx) =± √(2-k²) Now – k=√2 sin x k²=2sin²x (sinx – cosx) =± √(2-k²) = ± √(2-2sin²x) =± √2 (1-sin²x) = ± √2 cosx.