Question

Prove the trigonometric sum​

Answer 1

$HEYA \: \\ \\ GIVEN \: \: QUESTION \: \: Is \: \: \\ \sqrt{2} \cos( \frac{\pi}{4} + x) = \cos(x) - \sin(x) \\ \\ rhs \\ \\ \cos(x) - \sin( x) \\ Multiply \: \: Numerator \: \: and \ \\ Denominator \: \: by \: \sqrt{2} \: we \: have \\ \\ \ \ \sqrt{2} ( \frac{ \cos(x) }{ \sqrt{2} } - \frac{ \sin(x) }{ \sqrt{2} } ) \\ \\ \sqrt{2} ( \cos( \frac{\pi}{4} ) \cos(x) - \sin( \frac{\pi}{4} ) \sin(x) ) \\ \\ \sqrt{2} ( \cos( \frac{\pi}{4} + x) \: \: hence \: proved \: \: \\ \\ Note \: \: \: \: \: \: \: \\ \\ 1) \: \: \: \: \cos( \alpha - \beta ) = \cos( \alpha ) \cos( \beta ) + \sin( \alpha ) \sin( \beta ) \\ \\ 2) \: \: \: \sin( \frac{\pi}{4} ) = \cos( \frac{\pi}{4} ) = \frac{1}{ \sqrt{2} }$