Question

If $0<x<\frac{\pi }{4}$ and $\cos x+\sin x=\frac{5}{4},$ find the numerical values of $\cos x-\sin x.$

Answer 1

$cosx+sinx=\frac{5}{4}$

squaring on both sides

$(cosx+sinx{)}^2=\frac{25}{16}$

$co{s}^{2}x+2sinxcosx+si{n}^{2}x=\frac{25}{16}$

$1+2sinxcosx=\frac{25}{16}$

$2sinxcosx=\frac{25}{16}-1$

$2sinxcosx=\frac{25-16}{16}$

$2sinxcosx=\frac{9}{16}$

$(cosx-sinx{)}^2=(cosx-sinx{)}^2+4sinxcosx$

$(cosx-sinx{)}^2+4sinxcosx=\frac{25}{16}$

$(cosx+sinx{)}^2+\frac{18}{16}=\frac{25}{16}$

$(cosx-sinx{)}^2=\frac{25-18}{16}$

$cosx-sinx=\frac{\sqrt{7}}{4}$