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Question Number 11844 by minakshidahaval0202@gmail.co. last updated on 02/Apr/17

cotα+cosecα=k then find cosecα−cotα and also find cotα

$$\mathrm{cot}\alpha+\mathrm{cosec}\alpha={k}\:{then}\:{find}\:\mathrm{cosec}\alpha−{cot}\alpha\:{and}\:{also}\:{find}\:{cot}\alpha \\ $$

Answered by sma3l2996 last updated on 02/Apr/17

we have (i):cotα+cosecα=((cosα)/(sinα))+(1/(sinα))=k =((cosα+1)/(sinα))=(((cosα+1)(cosα−1))/(sinα(cosα−1)))=((−sin^2 α)/(sinα(cosα−1))) =(1/((1/(sinα))−((cosα)/(sinα))))=(1/(cosecα−cotα))=k (ii):cosecα−cotα=(1/k) let do (i)−(ii): 2cotα=k−(1/k) cotα=((k^2 −1)/k)

$${we}\:{have} \\ $$$$\left({i}\right):{cot}\alpha+{cosec}\alpha=\frac{{cos}\alpha}{{sin}\alpha}+\frac{\mathrm{1}}{{sin}\alpha}={k} \\ $$$$=\frac{{cos}\alpha+\mathrm{1}}{{sin}\alpha}=\frac{\left({cos}\alpha+\mathrm{1}\right)\left({cos}\alpha−\mathrm{1}\right)}{{sin}\alpha\left({cos}\alpha−\mathrm{1}\right)}=\frac{−{sin}^{\mathrm{2}} \alpha}{{sin}\alpha\left({cos}\alpha−\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{1}}{\frac{\mathrm{1}}{{sin}\alpha}−\frac{{cos}\alpha}{{sin}\alpha}}=\frac{\mathrm{1}}{{cosec}\alpha−{cot}\alpha}={k} \\ $$$$\left({ii}\right):{cosec}\alpha−{cot}\alpha=\frac{\mathrm{1}}{{k}} \\ $$$${let}\:{do} \\ $$$$\left({i}\right)−\left({ii}\right):\:\mathrm{2}{cot}\alpha={k}−\frac{\mathrm{1}}{{k}} \\ $$$${cot}\alpha=\frac{{k}^{\mathrm{2}} −\mathrm{1}}{{k}} \\ $$

Commented by sma3l2996 last updated on 02/Apr/17

I mean cotα=((k^2 −1)/(2k))

$${I}\:{mean}\:{cot}\alpha=\frac{{k}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{k}} \\ $$

Answered by ajfour last updated on 02/Apr/17

cosec^2 α − cot^2 α =1 (cosec α −cot α)(cosec α+cot α)=1 (cosec α − cot α )(k) =1 cosec α −cot α = (1/k) ...(ii) and as cosec α+cot α = k ...(i) (i)−(ii) gives 2cot α= k−(1/k) cot α= ((k^2 −1)/(2k)) .

$$\mathrm{cosec}\:^{\mathrm{2}} \alpha\:−\:\mathrm{cot}\:^{\mathrm{2}} \alpha\:=\mathrm{1} \\ $$$$\left(\mathrm{cosec}\:\alpha\:−\mathrm{cot}\:\alpha\right)\left(\mathrm{cosec}\:\alpha+\mathrm{cot}\:\alpha\right)=\mathrm{1} \\ $$$$\left(\mathrm{cosec}\:\alpha\:−\:\mathrm{cot}\:\alpha\:\right)\left({k}\right)\:=\mathrm{1} \\ $$$$\mathrm{cosec}\:\alpha\:−\mathrm{cot}\:\alpha\:=\:\frac{\mathrm{1}}{{k}}\:\:\:...\left({ii}\right) \\ $$$${and}\:{as}\:\:\mathrm{cosec}\:\alpha+\mathrm{cot}\:\alpha\:=\:{k}\:\:\:\:...\left({i}\right) \\ $$$$\left({i}\right)−\left({ii}\right)\:{gives} \\ $$$$\mathrm{2cot}\:\alpha=\:{k}−\frac{\mathrm{1}}{{k}} \\ $$$$\mathrm{cot}\:\alpha=\:\frac{{k}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{k}}\:. \\ $$

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