Question

If sin x + cos x = a, then
(i) ${sin}^6$x+${cos}^6$x = _______
(ii) |sin x - cos x| = _______.

Answer 1

We have $sinx+cosx=a$

so

$\left(i\right)si{n}^{6}x+co{s}^{6}x=$

by using $(a^3+b^3)$we get

$(si{n}^{2}x+co{s}^{2}x{)}^3-3si{n}^{2}x.co{s}^{2}x(si{n}^{2}x+co{s}^{2}x)$

$=1-3[\frac{(sinx+cosx{)}^2-(si{n}^{2}x+co{s}^{2}x)}{2}{]}^2$

$=1-\frac{3(a^2-1{)}^2}{4}$

$=\frac{[4-3(a^2-1\left)\right]}{4}$

(ii) $|sinx-cosx|$

$=\sqrt{(sinx-cosx{)}^2}$

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

$=\sqrt{a^2-4\left(\frac{a^2-1}{2}\right)}$

$=\sqrt{\frac{2a^2-4a^2+4}{2}}$

$=\sqrt{\frac{-2a^2+4}{2}}$

$=\sqrt{2-a^2}$