e)
[tex]L = sinxcosx(tgx + ctgx) = sinxcosx(\frac{sinx}{cosx} + \frac{cosx}{sinx}) = sin^2x+ cos^2x = 1 = P \\ L = P[/tex]
f)
[tex]L = (tgx + ctgx)^2 - (tgx - ctgx)^2= tg^2x + 2tgx \cdot ctgx + ctg^2x - (tg^2x -2tgx \cdot ctgx + \\ + ctg^2x) = tg^2x + 2 \cdot 1 + ctg^2x - (tg^2x -2 \cdot 1 + ctg^2x) =tg^2x + 2+ \\ + ctg^2x - tg^2x +2 - ctg^2x = 4 = P \\ L = P[/tex]
g)
[tex]L = \frac{1}{1+tg^2x} + \frac{1}{1+ctg^2x} = \frac{1+ctg^2x}{(1+tg^2x)(1+ctg^2x)} + \frac{1+tg^2x}{(1+tg^2x)(1+ctg^2x)} = \\ = \frac{1+ctg^2x+1+tg^2x}{1+ctg^2x+ tg^2x + tg^2xctg^2x} = \frac{tg^2x+ctg^2x+2}{tg^2x+ ctg^2x+ 1 + (tgxctgx)^2} = \frac{tg^2x+ctg^2x+2}{tg^2x+ ctg^2x+ 1 +1^2} = \\ = \frac{tg^2x+ctg^2x+2}{tg^2x+ ctg^2x+ 2} =1 = P \\ L = P[/tex]
