Ctg*x+tg*x=2.....* квадрат

[latex]ctg^2x+tg^2x=2;\\ D(f): \left \{ {{\cos^2x\neq0} \atop {\sin^2x\neq0}} \right.==> \left \{ {{\sin x\neq0} \atop {\cos x\neq0} \right. ==>x\neq \frac{\pi k}{2},\ k\in Z; \\ \frac{\cos^2x}{\sin^2x}+\frac{\sin^2x}{\cos^2x}-2=0;\\ \frac{\cos^2x}{\sin^2x}-1+\frac{\sin^2x}{\cos^2x}-1=0;\\ \frac{\cos^2x-sin^2x}{\sin^2x}+\frac{\sin^2x-\cos^2x}{\cos^2x}=0;\\ \frac{\cos2x}{\sin^2x}-\frac{\cos2x}{\cos^2x}=0;\\ \cos2x\left(\frac{1}{\sin^2x}-\frac{1}{\cos^2x}\right)=0;\\[/latex] [latex]\cos2x\cdot\frac{\cos^2x-\sin^2x}{\sin^2x\cos^2x}=0;\\ \frac{\cos^22x}{\sin^2x\cos^2x}=0;\\ \cos^22x=0;\\ \cos2x=0;\\ 2x=\frac{\pi }{2}+\pi n, n\in Z;\\ x=\frac{\pi}{4}+\frac{\pi n}{2}, n\in Z;\\ [/latex] [latex]tgx=t;\\ \frac1 t^2+t^2=2;\\ \frac{t^4-2t^2+1}{t^2}=0; (t^2-1)^2=0;\\ t^2-1=0;\\ t^2=1;\\ t=\pm1;\\ tg x=\pm1;\\ x=\frac{\pi}{4}+\frac{\pi n}{2} n\in Z[/latex]