Trzy tożsamości do sprawdzenia

[Kiepas]

a.) \(\displaystyle{ \cos^{4}x+\sin^{4}x=1-2\sin^{2}x\cos^{2}x}\)

b.) \(\displaystyle{ cos^{4}x-\sin^{4}x=\cos^{2}x-\sin^{2}x}\)

c.) \(\displaystyle{ (\tan x+\cot x)^{2}=\frac{1}{\sin^{2}x \cos^{2}x}}\)

[natkoza]

a)
\(\displaystyle{ cos^4+sin^4=(cos^2x)^2+(sin^2x)^2=(cos^2x+sin^2x)^2-2sin^2xcos^2x=1-2sn^2xcos^2x}\)
b)
\(\displaystyle{ cos^4x-sin^4x=(cos^2x)^2-(sin^2x)^2=(cos^2x+sin^2x)(cos^2x-sin^2x)=cos^2x-sin^2x}\)

[Szemek]

b)
\(\displaystyle{ L=\cos^4 x - \sin^4 x = (\sin^2 x + \cos^2 x)(\sin^2 x - \cos^2 x) = \sin^2 x - \cos^2 x = P}\)

[natkoza]

\(\displaystyle{ (tgx+ctgx)^2=(\frac{sinx}{cosx}+\frac{cosx}{sinx})^2=(\frac{sin^2+cos^2x}{sinxcosx})^2=(\frac{1}{sinxcosx})^2=\frac{1}{sin^2xcos^2x}}\)