If a1,a2,…,an are in A.P. with common difference d (where d=0 ); then the sum of the series sind(coseca1coseca2+coseca2coseca3+…+cosec an−1cosecan) is equal to cota1−cotan
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Step-by-Step Solution
Step 1: Analyze the general term
The given series involves terms of the form cosecakcosecak+1. We can rewrite this using the definition of cosecant as 1/(sinaksinak+1). This will help us simplify the expression.
Step 2: Introduce sind into the general term
Since a1,a2,…,an are in A.P. with common difference d, we know that ak+1−ak=d. We can substitute sind with sin(ak+1−ak) in the numerator. This is a key step to apply a trigonometric identity.
Step 3: Apply trigonometric identity
Using the trigonometric identity sin(A−B)=sinAcosB−cosAsinB, we expand the numerator. Then, we divide each term by the denominator sinaksinak+1. This simplifies to cotak−cotak+1, which is a telescoping form.
Step 4: Sum the series
Now we sum the series. Each term is of the form (cotak−cotak+1). This is a telescoping series, meaning that intermediate terms cancel out. The sum reduces to the first term of the first pair and the last term of the last pair.