Verified answer

1)

[tex] \tan {}^{2} ( \alpha ) + \tan( \alpha ) \cot( \alpha ) = \tan {}^{2} ( \alpha ) + \tan( \alpha ) \times \frac{1}{ \tan( \alpha ) } = \\ = \tan {}^{2} ( \alpha ) + 1 = \frac{ \sin {}^{2} ( \alpha ) }{ \cos {}^{2} ( \alpha ) } + \frac{ \cos {}^{2} ( \alpha ) }{ \cos {}^{2} ( \alpha ) } = \frac{ \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) }{ \cos {}^{2} ( \alpha ) } = \frac{1}{ \cos {}^{2} ( \alpha ) } [/tex]

2)

[tex]( \sin(x) + \cos(x) ) {}^{2} - 2\sin( x ) \cos(x) = \\ = \sin {}^{2} (x) + 2\sin(x) \cos(x) + \cos {}^{2} (x) - 2 \sin(x) \cos(x) = \\ = \sin {}^{2 } (x) + \cos {}^{2} (x) = 1[/tex]