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Question

# If x1,x2,x3,...,xn are in A.P. whose common difference is α, then the value of sinα(secx1secx2+secx2secx3+...+secxn−1+secxn) is

A
sin(n1)αcosx1cosxn
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B
sinnαcosx1cosxn
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C
sin(n1)αcosx1cosxn
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D
sinnαcosx1cosxn
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Solution

## The correct option is A sin(n−1)αcosx1cosxnGiven x1,x2,x3,........,xn are in A.P and, xn−xn−1=................=x3−x2=x2−x1=α Now,sinα(secx1.secx2+secx2.secx3+...........+secxn−1.secxn) =sin(x2−x1)cosx1.cosx2+sin(x3−x2)cosx2.cosx3+.......+sin(xn−xn−1)cosxn−1.cosxn=sinx2.cosx1−cosx2.sinx1)cosx1.cosx2+sinx3.cosx2−cosx3.sinx2)cosx2.cosx3+.......+sinxn.cosxn−1−cosxn.sinxn−1)cosxn−1.cosxn=sinxncosxn−sinx1cosx1=sin(xn−x1)cosx1.cosxn=sinα(n−1)cosx1.cosxnHence, option 'A' is correct.

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