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Question

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

A
sinθcosx1cosxn
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B
sin(n1)θcosx1cosxn
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C
sinnθcosx1cosxn
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D
cos(n1)θcosx1cosxn
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Solution

## The correct option is B sin(n−1)θcosx1cosxnGiven x1,x2,x3,........,xn are in A.P and, xn−xn−1=................=x3−x2=x2−x1=θ …(1)Now,sinθ(secx1.secx2+secx2.secx3+...........+secxn−1.secxn) =sin(θ)cosx1.cosx2+sin(θ)cosx2.cosx3+.......+sin(θ)cosxn−1.cosxn=sin(x2−x1)cosx1.cosx2+sin(x3−x2)cosx2.cosx3+.......+sin(xn−xn−1)cosxn−1.cosxn from (1)=sinx2.cosx1−cosx2.sinx1cosx1.cosx2+sinx3.cosx2−cosx3.sinx2cosx2.cosx3+.......+sinxn.cosxn−1−cosxn.sinxn−1cosxn−1.cosxn=sinx2cosx2−sinx1cosx1+sinx3cosx3−sinx2cosx2+sinx4cosx4−sinx3cosx3+............+sinxncosxn−sinx(n−1)cosx(n−1)=sinxncosxn−sinx1cosx1=sin(xn−x1)cosx1.cosxn=sinθ(n−1)cosx1.cosxnHence, option 'B' is correct.

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