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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)=
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)=
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Solution
The correct option is A sin(n−1)αcos x1 cos xn
We 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)}
We 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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