Question

Only legends can solve question number g
Salute u if u solved it
...
(TRIGONOMETRIC RATIOS OF SUB MULTIPLES ANGLES)

(SOLVE QUESTION NO.G)

Answer 1

$Answer \: \\ \\ Given \: \: Question \: \: \: Is \: \: \\ \\ \frac{ \sin {}^{3} ( \frac{x}{2} ) - \cos {}^{3} ( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} ) } = 1 + \frac{1}{2} \sin(x) \\ \\ lhs \\ \\ \frac{ \sin {}^{3} ( \frac{x}{2} ) - \cos {}^{3} ( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} ) } \\ \\ \frac{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )( \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )) {}^{2} + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} } \\ \\ \frac{( \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )) {}^{2} + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{ } \\ \\ \frac{ \sin {}^{2} ( \frac{x}{2} ) + \cos {}^{2} ( \frac{x}{2} ) - 2 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{} \\ \\ 1 + \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) \\ \\ 1 + \frac{2}{2} \frac{ \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{} \\ \\ 1 + \frac{1}{2} \sin( \frac{2x}{2} ) \\ \\ 1 + \frac{1}{2} \sin(x) \\ \\ Note \: \: \: \\ \\ 1) \: \: \alpha {}^{3} - \beta {}^{3} = \alpha - \beta ( \alpha - \beta ) {}^{2} + 3 \alpha \beta \\ \\ 2) \: \: \: \sin(2 \alpha ) = 2 \sin( \alpha ) \cos( \alpha )$

Answer 2

Step-by-step explanation:

$\begin{lgathered} \\ \frac{ \sin {}^{3} ( \frac{x}{2} ) - \cos {}^{3} ( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} ) } = 1 + \frac{1}{2} \sin(x) \\ \\ \\ LHS \\ \\ \\ \frac{ \sin {}^{3} ( \frac{x}{2} ) - \cos {}^{3} ( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} ) } \\ \\ \\ \frac{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )( \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )) {}^{2} + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{ \sin( \frac{x}{2} ) - \cos( \frac{x}{2} } \\ \\ \\ \frac{( \sin( \frac{x}{2} ) - \cos( \frac{x}{2} )) {}^{2} + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{ } \\ \\ \\ \frac{ \sin {}^{2} ( \frac{x}{2} ) + \cos {}^{2} ( \frac{x}{2} ) - 2 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) + 3 \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{} \\ \\ \\ 1 + \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) \\ \\ \\ 1 + \frac{2}{2} \frac{ \sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }{} \\ \\ \\ 1 + \frac{1}{2} \sin( \frac{2x}{2} ) \\ \\ \\ 1 + \frac{1}{2} \sin(x) \\ \\ \\ \end{lgathered}$