Question

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$tga \times ctga - \frac{sin {}^{2 } - 1 } {1 - cos {}^{2}a }$
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Answer 1

$\mathrm{tg}( \alpha ) \times \mathrm{ctg} ( \alpha ) - \frac{ \sin ^{2} ( \alpha ) - 1}{1 - \cos^{2} ( \alpha ) } = \frac{ \sin( \alpha ) }{ \cos( \alpha ) } \times \frac{ \cos( \alpha ) }{ \sin( \alpha ) } - \frac{ \sin ^{2} ( \alpha ) - 1}{1 - \cos^{2} ( \alpha ) } = 1 - \frac{ \sin ^{2} ( \alpha ) - 1}{1 - \cos^{2} ( \alpha ) } = 1 - \frac{ \sin ^{2} ( \alpha ) - 1}{ \sin^{2} ( \alpha ) } = 1 - \frac{\sin^{2} ( \alpha )}{\sin^{2} ( \alpha )} + \frac{1}{\sin^{2} ( \alpha )} = 1 - 1 + \frac{1}{\sin^{2} ( \alpha )} = \frac{1}{\sin^{2} ( \alpha )} = 1 + \mathrm{ctg^{2} }( \alpha )$