przekształcenie wzoru

[mat1989]

\(\displaystyle{ \frac{a}{\sqrt{a^2+h^2}}=tg }\)
ma wyjść : \(\displaystyle{ h=\frac{a}{sin\alpha}\sqrt{cos2\alpha}}\)

a ja licze tak:
\(\displaystyle{ \frac{a^2}{a^2+h^2}=tg^2\alpha\\a^2=tg^2\alpha(a^2+h^2)\\h^2=\frac{a^2(1-tg^2\alpha)}{tg^2\alpha}}\)
i jak sprawdzić czy to jest to samo?

[natkoza]

\(\displaystyle{ h^2=\frac{a^2(1-tg^2\alpha)}{tg^2\alpha}=\frac{a^2(1-\frac{sin^2\alpha}{cos^2\alpha})}{\frac{sin^2\alpha}{cos^2\alpha}}=\frac{a^2cos^2\alpha(1-\frac{sin^2\alpha}{cos^2\alpha})}{sin^2\alpha}=\frac{a^2cos^2\alpha-a^2sin^2\alpha}{sin^2\alpha}=\frac{a^2(cos^2\alpha-sin^2\alpha)}{sin^2\alpha}=\frac{a^2cos2\alpha}{sin^2\alpha}\\
h=\sqrt{\frac{a^2cos2\alpha}{sin^2\alpha}}=\frac{a\sqrt{cos2\alpha}}{sin\alpha}=\frac{a}{sin\alpha}\cdot \sqrt{cos2\alpha}}\)

[mat1989]

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