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Question Number 146044 by mathdanisur last updated on 10/Jul/21
Simplify:  ((sin^3 α)/(1-cosα)) + ((cos^3 α)/(sinα+1)) = ?
$${Simplify}: \\ $$$$\frac{{sin}^{\mathrm{3}} \alpha}{\mathrm{1}-{cos}\alpha}\:+\:\frac{{cos}^{\mathrm{3}} \alpha}{{sin}\alpha+\mathrm{1}}\:=\:? \\ $$
Answered by liberty last updated on 10/Jul/21
 ⇒ (((1−cos^2 α)sin α)/(1−cos α)) + (((1−sin^2 α)cos α)/(1+sin α))   = sin α(1+cos α)+cos α(1−sin α)  =sin α+cos α
$$\:\Rightarrow\:\frac{\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \alpha\right)\mathrm{sin}\:\alpha}{\mathrm{1}−\mathrm{cos}\:\alpha}\:+\:\frac{\left(\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \alpha\right)\mathrm{cos}\:\alpha}{\mathrm{1}+\mathrm{sin}\:\alpha}\: \\ $$$$=\:\mathrm{sin}\:\alpha\left(\mathrm{1}+\mathrm{cos}\:\alpha\right)+\mathrm{cos}\:\alpha\left(\mathrm{1}−\mathrm{sin}\:\alpha\right) \\ $$$$=\mathrm{sin}\:\alpha+\mathrm{cos}\:\alpha \\ $$
Commented by mathdanisur last updated on 10/Jul/21
thankyou Ser
$${thankyou}\:{Ser} \\ $$

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