Question

Вычислите значение функции y:
$y=sin^{4} x + cos^{4}x$ , если tg x = 2

Answer 1

$tg^2x + 1 = frac{sin^2x}{cos^2x} + 1 = frac{sin^2x + cos^2x}{cos^2x} = frac{1}{cos^2x} = textgreater cos^2x = frac{1}{1+tg^2x} cos^2x = 1 -sin^2x$
$1 - sin^2x = frac{1}{1+tg^2x} = textgreater$
$1 - frac{1}{1+tg^2x} = frac{1+tg^2x - 1}{1 + tg^2x} = frac{tg^2x}{1+ tg^2x} = sin^2x y = sin^4x + cos^4x = (frac{tg^2x}{1+ tg^2x})^2 + (frac{1}{1+tg^2x})^2 = (frac{4}{5})^2 + ( frac{1}{5})^2 = frac{17}{25}$