If `cos(x-y),cos x` and `cos(x+y)` are in HP, then `cos x sec ((y)/(2))=`

Options

A. `pm sqrt(2)`
B. `pm1//sqrt(2)`
C. `pm2`
D. none of these

Answer

A

Explanation

cos (x-y),cos x, cos (x+y) are in H.P.
`rArr" "cosx=(2cos(x-y)cos(x+y))/(cos(x-y)+cos(x+y))`
`rArr" "cosx=(2(cos^(2)x-sin^(2)y))/(2cosxcosy)`
`rArr" "cos^(2)xcosy=cos^(2)x-sin^(2)y`
`rArr" "cos^(2)x(1-cosy)=sin^(2)y`
`rArr" "cos^(2)x(1-cosy)=(1-cos^(2)y)`
`rArr" "cos^(2)x(1-cosy)=(1+cosy)(1-cosy)`
`rArr" "cos^(2)x=1+cosy`
`rArr" "cos^(2)x=2cos^(2)y//2`
`rArr" "cos^(2)xsec^(2)y//2=2`
`rArr" "cosxsec(y//2)=pmsqrt(2)`