Question

Доказать тождество:
ctgx/ctgx+tgx=cos^2x

Answer 1

(1+sinx+cosx)/(1+sinx-cosx)=ctg(x/2);

(1+sin(π/2-x)+sinx)/(1+sinx-sin(π/2-x))=(1+2*sin(π/4)*cos(π/4-x))/(1+2*sin(x-π/4)*cos(π/4))=(1+√2*cos(π/4-x))/(1+√2*sin(π/4-x))=ctg(x/2). - доказанно.

Answer 2

$\frac{Ctgx}{Ctgx+tgx}= \frac{Ctgx}{Ctgx+ \frac{1}{Ctgx} } = \frac{Ctg ^{2}x }{1+Ctg ^{2}x }= \frac{ \frac{Cos ^{2}x }{Sin ^{2} x} }{1+ \frac{Cos ^{2} x}{Sin ^{2} x} } = \frac{ \frac{Cos ^{2} x}{Sin ^{2} x} }{ \frac{Sin ^{2} x+Cos ^{2} x}{Sin ^{2} x} }= \frac{ \frac{Cos ^{2} x}{Sin ^{2} x} }{ \frac{1}{Sin ^{2} x} }=$$\frac{Cos ^{2} x*Sin ^{2} x}{Sin ^{2} x}=Cos ^{2}x\\\\Cos ^{2} x=Cos ^{2} x$
тождество доказано