Question

if sinx + cosx = 1/2, then find sin^4 x + cos^4 x as a rational number of equals

Answer 1

$\sin x + cos x = \frac{1}{2} \\ on \: squaring \: we \: get \\ {(sin x + cos x) }^{2} = {( \frac{1}{2} )}^{2} \\ { \sin }^{2} x + { \cos }^{2} x + 2sinxcosx = \frac{1}{4} \\ 1 + 2sinxcosx = \frac{1}{4} \\ 2sinxcosx = \frac{1}{4} - 1 \\ 2sinxcosx = - \frac{3}{4} \\ sinxcosx = - \frac{3}{8} \\ \\ now \\ \\ { \sin }^{2} x + { \cos }^{2} x = 1 \\ squaring \: we \: get \\ { \sin }^{4} x + { \cos }^{4} x + 2{ \sin }^{2} x { \cos }^{2} x = {1}^{2} \\ { \sin }^{4} x + { \cos }^{4} x + 2( - \frac{3}{8} )= {1}^{2} \\ { \sin }^{4} x + { \cos }^{4} x - \frac{3}{4} = 1 \\ { \sin }^{4} x + { \cos }^{4} x = 1 + \frac{3}{4} \\{ \sin }^{4} x + { \cos }^{4} x = \frac{7}{4}$