$a)\frac{\mathrm{tg}x+1}{\mathrm{ctg}x+1}=\frac{\frac{\sin x}{\cos x}+1}{\frac{\cos x}{\sin x}+1}=\frac{\frac{\sin x}{\cos x}+\frac{\cos x}{\cos x}}{\frac{\cos x}{\sin x}+\frac{\sin x}{\sin x}}=\frac{\frac{\sin x+\cos x}{\cos x}}{\frac{\sin x+\cos x}{\sin x}}=\frac{\sin x+\cos x}{\cos x}\cdot \frac{\sin x}{\sin x+\cos x}=\frac{\sin x}{\cos x}=\mathrm{tg}x$  

$L=P$  

 

$b)\frac{{\sin }^{2}x-{\cos }^{2}x}{\sin x\cos x}=\frac{{\sin }^{2}x}{\sin x\cos x}-\frac{{\cos }^{2}x}{\sin x\cos x}=\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}=\mathrm{tg}x-\mathrm{ctg}x$  

$L=P$  

 

$c){\left(\mathrm{tg}x+1\right)}^2={\mathrm{tg}}^{2}x+2\mathrm{tg}x+1=\frac{{\sin }^{2}x}{{\cos }^{2}x}+2\mathrm{tg}x+\frac{{\cos }^{2}x}{{\cos }^{2}x}=\frac{{\sin }^{2}x+{\cos }^{2}x}{{\cos }^{2}x}+2\mathrm{tg}x=\frac{1}{{\cos }^{2}x}+2\mathrm{tg}x$  

$L=P$  

 

$d)\frac{\cos x}{1+\mathrm{tg}x}+\frac{\sin x}{1+\mathrm{ctg}x}=\frac{\cos x}{\frac{\cos x}{\cos x}+\frac{\sin x}{\cos x}}+\frac{\sin x}{\frac{\sin x}{\sin x}+\frac{\cos x}{\sin x}}=\frac{\cos x}{\frac{\cos x+\sin x}{\cos x}}+\frac{\sin x}{\frac{\sin x+\cos x}{\sin x}}=\cos x\cdot \frac{\cos x}{\sin x+\cos x}+\sin x\cdot \frac{\sin x}{\sin x+\cos x}=$  

$=\frac{{\cos }^{2}x}{\sin x+\cos x}+\frac{{\sin }^{2}x}{\sin x+\cos x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{\sin x+\cos x}=\frac{1}{\sin x+\cos x}$  

$L=P$