Is it possible to prove that sin d(csc a1 csca2 +csc a2 csc a3+ ....+csc an−1csc an)=cot a1−cot an ,if a1,a2,a3,……..an are in A.P with common difference d(d≠0)

Question

Is it possible to  prove that sin d(csc a1 csca2 +csc a2 csc a3+ ....+csc an−1csc an)=cot a1−cot an ,if a1,a2,a3,……..an are in A.P with common difference d(d≠0)

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We are given that a1,a2,a3,……..an are in A.P with common difference d(d≠0)We are asked to prove that sin d(csc a1 csca2 +csc a2 csc a3+ ....+csc an−1csc an)=cot a1−cot anLet us assume that the LHS of given result as⇒L= sin d(csc a1 csca2 +csc a2 csc a3+ ....+csc an−1csc an)We know that the formula of cosecant function as cscθ=1sinθBy using the above formula and multiplying the outside term to inside the bracket in LHS we get⇒L=sin d(1sina1sina2+1sina2sina3+...+1sinan-1sinan)⇒L=sindsina1sina2+sindsina2sina3+...+sindsinan-1sinanWe know that the condition that the common difference of an A.P is given as the difference of any two consecutive terms in the A.P that is if a1,a2,a3,……..an  are in A.P with common difference d then⇒d=a2−a1=a3-a2=....=an−an−1Now, let us use the required values in the required terms of the above equation then we get⇒L= sin⁡(a2-a1)sina1sina2+sin(a3-a2)sina2sina3+...+sin⁡(an-an-1)sinan-1sinanWe know that the formula of composite angle for sine function assin(A−B)=sinAcosB−cosAsinBBy using this formula of composite angle to the above equation we get ⇒L= = sina2cosa1-cosa2sina1sina1sina2+sina3cosa2-cosa3sina2)sina2sina3+...+ sinancosan-1-cosansinan-1)sinan-1sinanNow, let us divide the each term to individual terms then we get⇒L=cota1−cota2+cota2−cota3+cota3−cota4+...+cotan-1−cotan⇒L=cota1−cotanHere we can see that the value of LHS is equal to RHSTherefore we can conclude that the required result has been proved that is∴ sin d(csc a1 csca2 +csc a2 csc a3+ ....+csc an−1csc an)=cot a1−cot an

Is it possible to prove that sin d(csc $a_{1}$ csc $a_{2}$ +csc $a_{2}$ csc $a_{3}$ + ....+csc $a_{n}$ −1csc $a_{n}$ )=cot $a_{1}$ −cot $a_{n}$ ,if $a_{1}, a_{2}, a_{3}, \dots \dots . . a_{n}$ are in A.P with common difference d(d≠0)

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