Question

Can you please work out the sum ​

Answer 1

Answer:

$\frac{1 - \sin(θ). \cos(θ) }{ \cos(θ) ( \sec(θ) - cosec(θ)} \times \frac{ \sin {}^{2} (θ) - \cos {}^{2} (θ) }{ \sin {}^{3} (θ) + \cos {}^{3} (θ) }$

[ ∵ secθ=1/cosθ, & cosecθ=1/sinθ ]

[ ∵ a²–b²=(a+b)(a–b) ], &

[ ∵ a³+b³=(a+b)(a²–ab+b²) ]

$\frac{1 - \sin(θ) \cos(θ) }{ \cos(θ)( \frac{1}{ \cos(θ) } - \frac{1}{ \sin(θ) } )} \times \frac{ (\sin(θ) + \cos(θ))( \sin(θ) - \cos(θ)) }{( \sin(θ) + \cos(θ))( \sin {}^{2} (θ) + \cos {}^{2} (θ) - \sin(θ) \cos(θ) ) }$

[ ∵ sin²θ+cos²θ=1 ]

$\frac{(1 - \sin( θ) \cos(θ) ) }{ \frac{ \sin(θ) - \cos(θ) }{ \cos(θ) } } \times \frac{ \sin(θ) - \cos(θ) }{(1 - \sin(θ) \cos(θ )) }$

$\frac{ \sin(θ) }{( \sin(θ) - \cos(θ)) } \times {( \sin(θ) - \cos(θ) ) }$

$⇒ \sin(θ)$