If x 1 , x 2 , x 3 , . . . . . . , x n are in A.P. whose common difference is a, then the value of sin α (sec x 1 sec x 2 + sec x 2 sec x 3 +....... + sec x n − 1 , sec x n )= Options: A . &nbsp sin ( n − 1 ) α cos x 1 cos x n B . &nbsp sin n α cos x 1 cos x n C . &nbsp sin (n - 1) α cos x 1 cos x n D . &nbsp sin n α cosx 1 cosx n

Answer: Option A : A (a) We have sin α sec x 1 sec x 2 + sin α sec x 2 sec x 3 +....... + sin α sec x n − 1 , sec x n ) = sin ( x 2 − x 1 ) cos x 1 cos x 2 + sin ( x 3 − x 2 ) cos x 2 cos x 3 + ........ + sin ( x n − x n − 1 ) cos x n − 1 cos x n = tan x 2 - tan x 1 + tan x 3 - tan x 2 + .......... + tan x n - tan x n − = tan x n - tan x 1 = sin ( x n − x n − 1 ) cos x n − 1 cos x 1 = sin ( n − 1 ) α cos x n cos x 1 {( ∵ x n = x 1 + (n - 1) α )}

If x1,x2,x3,......,xn are in A.P. whose common difference is a, then the value

Question

If $x_{1}, x_{2}, x_{3}, . . . . . ., x_{n}$ are in A.P. whose common difference is a, then the value of sin $α$ (sec $x_{1}$ sec $x_{2}$ + sec $x_{2}$ sec $x_{3}$ +....... + sec $x_{n - 1}$ , sec $x_{n}$ )=

Options:

A .

\frac{sin (n - 1) α}{cos x_{1} cos x_{n}}

B .

\frac{sin n α}{cos x_{1} cos x_{n}}

C . sin (n - 1)

α

cos x

_{1}

cos x

_{n}

D . sin n

α

cosx

_{1}

cosx

_{n}

Answer: Option A
:
A

(a) We have sin $α$ sec $x_{1}$ sec $x_{2}$ + sin $α$ sec $x_{2}$ sec $x_{3}$ +....... + sin $α$ sec $x_{n - 1}$ , sec $x_{n}$ )
= $\frac{sin (x_{2} - x_{1})}{cos x_{1} cos x_{2}}$ + $\frac{sin (x_{3} - x_{2})}{cos x_{2} cos x_{3}}$ + ........ + $\frac{sin (x_{n} - x_{n - 1})}{cos x_{n - 1} cos x_{n}}$
   = tan x $_{2}$ - tan x $_{1}$ + tan x $_{3}$ - tan x $_{2}$ + .......... + tan x $_{n}$ - tan x $_{n -}$
   = tan x $_{n}$ - tan x $_{1}$ = $\frac{sin (x_{n} - x_{n - 1})}{cos x_{n - 1} cos x_{1}}$ = $\frac{sin (n - 1) α}{cos x_{n} cos x_{1}}$ {( $∵$ x $_{n}$ = x $_{1}$ + (n - 1) $α$ )}