Question

Упростите выражение $frac{cos a}{1+sin a} + frac{1+sin a}{cos a}$ и найдите его значение при $a = - frac{ pi }{4}$

Answer 1

$frac{cos alpha }{1+sin alpha } + frac{1+sin alpha }{cos alpha} = frac{cos^2 alpha +(1+sin alpha )^2}{(1+sin alpha )cos alpha } =frac{cos^2 alpha +1+2sin alpha+sin^2 alpha }{(1+sin alpha )cos alpha } =\ =frac{2+2sin alpha }{(1+sin alpha )cos alpha } =frac{2(1+sin alpha) }{(1+sin alpha )cos alpha } = frac{2}{cos alpha }$
$alpha =- frac{ pi }{4} Rightarrow frac{2}{cos(- frac{ pi }{4} )} = frac{2}{ frac{ sqrt{2} }{2} } = frac{4}{sqrt{2} } =frac{4 sqrt{2} }{ 2}=2 sqrt{2}$