Question

If sin x + cos x = a, find the value of |sin x - cos x|.

Answer 1

sin x + cos x = a

squaring on both sides gives

$si{n}^{2}x+co{s}^{2}x+2sinxcosx=a^2$

$1+sin2x=a^2$

$sin2x=a^2-1$

$Ifsin2x=a^2-1\text{then}$

$(sinx-cosx{)}^2=si{n}^{2}x+co{s}^{2}x-2sinxcos$

$x=1-(a^2-1)=2-a^2$

Take square root on both sides, we get

$|sinx-cosx|=\sqrt{2-a^2}$