Question

(0)
cos (90° - A) sin (90° – A)/ tan (90 degree-A)
= 1 - cos2 A
​

Answer 1

$\frac{ \cos(90 - a) \sin(90 - a) }{ \tan(90 - a) } \\ remember \: that \\ \cos(90 - a) = \sin(a) \\ \sin(90 - a) = \cos(a) \: \\ \tan(90 - a) = \cot(a) \\ \cot(a) = \frac{ \cos(a) }{ \sin(a) } \\ so \: \frac{ \cos(90 - a) \sin(90 - a) }{ \tan(90 - a) } \\ = \frac{ \sin(a) \cos(a) }{ \cot(a) } \\ = \frac{ \sin(a) \cos(a) }{ \frac{ \cos(a) }{ \sin(a) } } \\ = \sin(a) \cos(a) \times \frac{ \sin(a) }{ \cos(a) } \\ = \sin(a) \times \sin(a) \\ = {sin}^{2} a \\ we \: know \: that \: \\ {sin}^{2} a + {cos}^{2} a = 1 \\ from \: this \\ {sin}^{2} a = 1 - {cos}^{2} a \\ hence \: proved$

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