Question

Daca x apartine ( 0, 90 ) si six(x) - cos(x) = 1 / 2
Calaculati
Sin2x
Cos2x

Answer 1

$sin(x)-cos(x)=\frac{1}{2} => (sin(x)-cos(x))^{2}=\frac{1}{4} => sin^{2}x-2sin(x)cos(x)+cos^2{x}=\frac{1}{4}\\dar sin^{2}x+cos^{2}x=1, => 1-2sinxcosx=\frac{1}{4}=>2sinxcosx=1-\frac{1}{4}=\frac{3}{4}\\dar sin(2x)=2sinxcosx\\deci sin(2x)=\frac{3}{4}\\|cos2x|=\sqrt{1-\frac{9}{16} }=\sqrt{\frac{7}{16}}$

=>cos(2x)=±$\sqrt{\frac{7}{16}}$