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Question

# If x1,x2,x3,....,xn are in A.P. whose common difference is α, then the value of sin α(secx1 secx2+secx2 secx3 +... ...+secxn−1 secxn)=

A
sin(n1)αcos x1 cos xn
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B
sin n αcos x1 cos xn
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C
sin(n1)α cos x1 cosxn
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D
sin n α cos x1 cosxn
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Solution

## The correct option is A sin(n−1)αcos x1 cos xnWe have sin α sec x1 sec x2+sin α sec x2 sec x3+.... ...+sin α sec xn−1sec xn =sin(x2−x1)cos x1 cos x2+sin(x3−x2)cos x2 cos x3+...+sin(xn−xn−1)cos xn−1 cos xn=tan x2−tan x1+tan x3−tan x2+...+tan xn−tan xn−1=tan xn−tan x1=sin(xn−x1)cos xn cos x1=sin(n−1)αcos xn cos x1 {(∵ xn=x1+(n−1)α)}

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