a)
[tex]L = \frac{1}{1-cosx} + \frac{1}{1+cosx} = \frac{1+cosx}{(1-cosx)(1+cosx)} + \frac{1-cosx}{(1-cosx)(1+cosx)} =\frac{1+cosx+1-cosx}{(1-cosx)(1+cosx)}= \\ = \frac{2}{1-cos^2x}= \frac{2}{sin^2x}=P \\ L = P[/tex]
b)
[tex]L = tg^2x - sin^2x= \frac{sin^2x}{cos^2x} - \frac{sin^2xcos^2x}{cos^2x} =\frac{sin^2x -sin^2xcos^2x}{cos^2x} =\frac{sin^2x(1 -cos^2x)}{cos^2x} = \\ = \frac{sin^2xsin^2x}{cos^2x} =\frac{sin^2x}{cos^2x} \cdot sin^2x = tg^2x \cdot sin^2x = P \\ L =P[/tex]
c)
[tex]L =\frac{tgx}{1-tg^2x} \cdot \frac{ctg^2x-1}{ctgx}=\frac{tgx}{tgxctgx-tg^2x} \cdot \frac{ctg^2x-tgxctgx}{ctgx}=\frac{tgx}{tgx(ctgx-tgx)} \cdot \frac{ctgx(ctgx-tgx)}{ctgx}=\\ = \frac{1}{ctgx-tgx} \cdot \frac{ctgx-tgx}{1}= \frac{ctgx-tgx}{ctgx-tgx}= 1 = P \\ L = P[/tex]
d)
[tex]L = \dfrac{sin^2x}{sinx - cosx} +\dfrac{sinx +cosx}{1-tg^2x} = \dfrac{sin^2x}{sinx - cosx} +\dfrac{sinx +cosx}{\frac{cos^2}{cos^2x} -\frac{sin^2}{cos^2x}} = \\ = \dfrac{sin^2x}{sinx - cosx} +\dfrac{sinx +cosx}{\frac{cos^2x-sin^2x}{cos^2x}} =\dfrac{sin^2x}{sinx - cosx} +\dfrac{cos^2x(sinx +cosx)}{cos^2x-sin^2x} =[/tex]
[tex]=\dfrac{sin^2x}{sinx - cosx} +\dfrac{cos^2x(sinx +cosx)}{(cosx-sinx)(cosx+sinx)} =\dfrac{sin^2x}{sinx - cosx} +\dfrac{cos^2x}{cosx-sinx} = \\ = \dfrac{sin^2x}{sinx - cosx} +\dfrac{cos^2x}{-(sinx-cosx)} = \dfrac{sin^2x}{sinx - cosx} -\dfrac{cos^2x}{sinx-cosx} = \\ = \dfrac{sin^2x-cos^2x}{sinx - cosx} = \dfrac{(sinx - cosx)(sinx +cosx)}{sinx - cosx} = sinx +cosx = P \\ L = P[/tex]
