Tożsamości trygonometryczne

[mikrobart]

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

2.
\(\displaystyle{ ctgx+ \frac{sinx}{1+cosx} = \frac{1}{sinx}}\)

3.
\(\displaystyle{ \frac{tgx}{tgx+ctgx} =sin^2x}\)

4.
\(\displaystyle{ \frac{ctgx}{ctgx+tgx} =cos^2x}\)

[bakala12]

1. Skorzystaj ze wzoru skróconego mnożenia \(\displaystyle{ a ^{2}-b ^{2}=(a+b)(a-b)}\) i z jedynki trygonometrycznej

[JakimPL]

2. \(\displaystyle{ \ctg x+ \frac{\sin x}{1+\cos x} = \frac{1}{\sin x}}\)

Klasycznie:

\(\displaystyle{ L = \ctg x+ \frac{\sin x}{1+\cos x} = \frac{\cos x}{\sin x}+\frac{\sin x}{1+\cos x} = \frac{\cos x(1 + \cos x)}{\sin x (1 + \cos x)}+\frac{\sin^2 x}{\sin x(1+\cos x)}=\frac{\sin^2 x+\cos^2 x +\cos x}{\sin x(1+\cos x)}=\frac{1 +\cos x}{\sin x(1+\cos x)}=\frac{1}{\sin x}=P}\)

4. \(\displaystyle{ \frac{\ctg x}{\ctg x+\tg x} =\cos^2x}\)

\(\displaystyle{ L=\frac{\ctg x}{\ctg x+\tg x} =\frac{\ctg x}{\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}}=\frac{\ctg x}{\frac{\cos^2 x + \sin^2 x}{\sin x\cos x}}=\frac{\ctg x \sin x\cos x}{\cos^2 x + \sin^2 x}=\ctg x \sin x\cos x = \frac{\cos x}{\sin x} \sin x\cos x = \cos^2 x = P}\)

[bakala12]

4.
\(\displaystyle{ L= \frac{ctgx}{ctgx+tgx} = \frac{ \frac{cosx}{sinx} }{ \frac{cosx}{sinx}+ \frac{sinx}{cosx} } = \frac{ \frac{cosx}{sinx} }{ \frac{cos ^{2}+sin ^{2} }{sinxcosx} } = \frac{cosx}{sinx} \cdot sinxcosx=cos ^{2}x=P}\)
3 analogicznie