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Simplify-1-1-cos-a-1-1-cos-a-and-leave-the-answer-in-the-form-sin-a-




Question Number 27580 by Mcfaithal last updated on 10/Jan/18
Simplify (1/(1−cos a)) + (1/(1+cos a))  and leave the answer in the   form⇒   sin a
$$\mathrm{Simplify}\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\:\mathrm{a}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}\:\mathrm{a}} \\ $$$$\mathrm{and}\:\mathrm{leave}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{the}\: \\ $$$$\mathrm{form}\Rightarrow\:\:\:\mathrm{sin}\:\mathrm{a} \\ $$
Answered by Joel578 last updated on 10/Jan/18
(1/(1 − cos a)) + (1/(1 + cos a)) = (((1 + cos a) + (1 − cos a))/((1 − cos a)(1 + cos a)))                                      = (2/(1 − cos^2  a))                                      = (2/(sin^2  a))
$$\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{cos}\:{a}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{cos}\:{a}}\:=\:\frac{\left(\mathrm{1}\:+\:\mathrm{cos}\:{a}\right)\:+\:\left(\mathrm{1}\:−\:\mathrm{cos}\:{a}\right)}{\left(\mathrm{1}\:−\:\mathrm{cos}\:{a}\right)\left(\mathrm{1}\:+\:\mathrm{cos}\:{a}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2}}{\mathrm{1}\:−\:\mathrm{cos}^{\mathrm{2}} \:{a}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2}}{\mathrm{sin}^{\mathrm{2}} \:{a}}\: \\ $$

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