Question

If `sec(theta+alpha)+sec(theta-alpha)=2sec(theta)`,prove that `cos(theta)=sqrt2 cos(alpha/2)`?

Answer 1

Please see below.

Explanation:

`sec(theta+alpha)+sec(theta-alpha)=2sec(theta)` can be written as

`1/cos(theta+alpha)+1/cos(theta-alpha)=2/costheta`

or `(cos(theta+alpha)+cos(theta-alpha))/(cos(theta+alpha)cos(theta-alpha))=2/cos(theta)`

or`(2xx2costhetacosalpha)/(cos(theta+alpha+theta-alpha)+cos(theta+alpha-theta+alpha))=2/costheta`

or `(2costhetacosalpha)/(cos2theta+cos2alpha)=1/costheta`

or `2cos^2thetacosalpha=cos2theta+cos2alpha`

or `2cos^2thetacosalpha=2cos^2theta-1+2cos^2alpha-1`

or `2cos^2theta(cosalpha-1)=2cos^2alpha-2`

or `cos^2theta=(cos^2alpha-1)/(cosalpha-1)=cosalpha+1`

or `cos^2theta=2cos^2(alpha/2)`

and `costheta=sqrt2cos(alpha/2)`