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If f(x)=(2^x+2^(-x))/2 , then f(x+y)f(x-...

If `f(x)=(2^x+2^(-x))/2` , then `f(x+y)f(x-y)` is equals to `1/2{f(2x)+f(2y)}` (b) `1/2{f(2x)-f(2y)}` (c) `1/4{f(2x)+f(2y)}` `1/4{f(2x)-f(2y)}`

A

` ( 1 ) /( 4 ) [f ( 2x ) - f ( 2y ) ] `

B

` ( 1 ) /( 2 ) [ f ( 2x ) - f ( 2y ) ] `

C

` ( 1 )/ ( 4 ) [ f ( 2x ) + f ( 2y ) ] `

D

` ( 1 ) /( 2 ) [ f ( 2x ) + f ( 2y ) ] `

Text Solution

Verified by Experts

The correct Answer is:
D
Given ` f ( x ) = ( 2 ^ x + 2 ^( - x ) ) /( 2 ) `
Now, ` f (x + y ) = ( 2 ^ ( x + y ) + 2^( -x - y ) ) / ( 2 ) `
and ` f ( x- y ) = ( 2^ ( x - y ) + 2 ^( - x + y ) ) /( 2 ) `
` therefore f (x + y ) * f ( x - y ) `
` = ( ( 2 ^ ( x + y ) + 2 ^( - x - y ) ) ) /( 2 ) * ( ( 2^ ( x- y ) + 2 ^( - x + y )))/( 2 ) `
= ` ( 2 ^ ( 2x ) + 2 ^( -2 y ) + 2^( 2y ) + 2 ^( -2x )) /( 4 ) `
` = ( 1 ) /( 2 ) { ((2^ ( 2x ) + 2 ^ ( -2 x ) ) /( 2 ) ) + ( ( 2 ^ ( 2y ) + 2 ^( - 2y ) ) /( 2 ) ) } `
` = ( 1 ) / ( 2 ) { f ( 2x ) + f ( 2y ) } `
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