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Question
If x1,x2,x3,......,xn are in A.P. whose common difference is a, then the value of sin α(sec x1 sec x2 + sec x2 sec x3 +....... + sec xn−1, sec xn)=
If x1,x2,x3,......,xn are in A.P. whose common difference is a, then the value of sin α(sec x1 sec x2 + sec x2 sec x3 +....... + sec xn−1, sec xn)=
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Solution
The correct option is A
(a) We have sin α sec x1 sec x2 + sin α sec x2 sec x3 +....... + sin α sec xn−1, sec xn)
=sin(x2−x1)cosx1 cosx2 + sin(x3−x2)cosx2 cosx3 + ........ + sin(xn−xn−1)cosxn−1 cosxn
= tan x2 - tan x1 + tan x3 - tan x2 + .......... + tan xn - tan xn−
= tan xn - tan x1 = sin(xn−xn−1)cosxn−1 cosx1 = sin(n−1)αcosxn cosx1 {(∵ xn = x1 + (n - 1)α)}
(a) We have sin α sec x1 sec x2 + sin α sec x2 sec x3 +....... + sin α sec xn−1, sec xn)
=sin(x2−x1)cosx1 cosx2 + sin(x3−x2)cosx2 cosx3 + ........ + sin(xn−xn−1)cosxn−1 cosxn
= tan x2 - tan x1 + tan x3 - tan x2 + .......... + tan xn - tan xn−
= tan xn - tan x1 = sin(xn−xn−1)cosxn−1 cosx1 = sin(n−1)αcosxn cosx1 {(∵ xn = x1 + (n - 1)α)}
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