zadanie 1   Udowodnij tożsamość trygonometryczną.   a) sinx/1-cosx + 1-cosx/sinx=2/sinx   b) ctgx-tgx/sinx+cosx= 1/sinx-1/cosx   c) tgx-ctgx/tgx+ctgx= tg^2 x-1/tg^2 x+1    

a) [latex] frac{sin x}{1-cos x} + frac{1-cos x}{sin x} = frac{2}{sin x} \ frac{sin^2x+(1-cos x)^2}{(1-cos x)sin x} = frac{2}{sin x} \ frac{sin^2x+1-2cos x+cos^2x}{(1-cos x)sin x} = frac{2}{sin x} \ frac{1+1-2cos x}{(1-cos x)sin x} = frac{2}{sin x} \ frac{2(1-cos x)}{(1-cos x)sin x} = frac{2}{sin x} \ frac{2}{sin x} = frac{2}{sin x} \[/latex] b) [latex] frac{cot x - an x}{sin x +cos x} = frac{1}{sin x}- frac{1}{cos x} \ frac{ frac{cos x}{sin x} - frac{sin x}{cos x} }{sin x +cos x} = frac{cos x - sin x}{sin x cos x}} \ frac{ frac{cos^2 x-sin^2 x}{sin x cos x} }{sin x +cos x} = frac{cos x - sin x}{sin x cos x}} \ frac{(cos x-sin x)(cos x+sin x)}{(sin x +cos x)sin x cos x} = frac{cos x - sin x}{sin x cos x}} \ frac{cos x - sin x}{sin x cos x}} = frac{cos x - sin x}{sin x cos x}} [/latex] c) [latex] frac{ an x- cot x}{ an x+ cot x} = frac{ an^2x-1}{ an^2x+1} \ frac{ an x- frac{1}{ an x} }{ an x+ frac{1}{ an x} } = frac{ an^2x-1}{ an^2x+1} \ frac{frac{ an ^2x-1}{ an x} }{frac{ an ^2x+1}{ an x} } = frac{ an^2x-1}{ an^2x+1} \ frac{ an ^2x-1}{ an x} {frac{ an x}{ an ^2x+1} = frac{ an^2x-1}{ an^2x+1} \ [/latex] [latex]frac{ an^2x-1}{ an^2x+1} =frac{ an^2x-1}{ an^2x+1} [/latex]