If $sec (θ + α) + sec (θ - α) = 2 sec (θ)$ $sec(theta+alpha)+sec(theta-alpha)=2sec(theta)$ ,prove that $cos (θ) = \sqrt{2} cos (\frac{α}{2})$ $cos(theta)=sqrt2 cos(alpha/2)$ ?

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If $sec (θ + α) + sec (θ - α) = 2 sec (θ)$ $sec(theta+alpha)+sec(theta-alpha)=2sec(theta)$ ,prove that $cos (θ) = \sqrt{2} cos (\frac{α}{2})$ $cos(theta)=sqrt2 cos(alpha/2)$ ?

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Answer 1 · 2018-02-11T16:59:59

Please see below.

Explanation:

$sec (θ + α) + sec (θ - α) = 2 sec (θ)$ $sec(theta+alpha)+sec(theta-alpha)=2sec(theta)$ can be written as

$\frac{1}{cos (θ + α)} + \frac{1}{cos (θ - α)} = \frac{2}{cos θ}$ $1/cos(theta+alpha)+1/cos(theta-alpha)=2/costheta$

or $\frac{cos (θ + α) + cos (θ - α)}{cos (θ + α) cos (θ - α)} = \frac{2}{cos (θ)}$ $(cos(theta+alpha)+cos(theta-alpha))/(cos(theta+alpha)cos(theta-alpha))=2/cos(theta)$

or $\frac{2 \times 2 cos θ cos α}{cos (θ + α + θ - α) + cos (θ + α - θ + α)} = \frac{2}{cos θ}$ $(2xx2costhetacosalpha)/(cos(theta+alpha+theta-alpha)+cos(theta+alpha-theta+alpha))=2/costheta$

or $\frac{2 cos θ cos α}{cos 2 θ + cos 2 α} = \frac{1}{cos θ}$ $(2costhetacosalpha)/(cos2theta+cos2alpha)=1/costheta$

or $2 {cos}^{2} θ cos α = cos 2 θ + cos 2 α$ $2cos^2thetacosalpha=cos2theta+cos2alpha$

or $2 {cos}^{2} θ cos α = 2 {cos}^{2} θ - 1 + 2 {cos}^{2} α - 1$ $2cos^2thetacosalpha=2cos^2theta-1+2cos^2alpha-1$

or $2 {cos}^{2} θ (cos α - 1) = 2 {cos}^{2} α - 2$ $2cos^2theta(cosalpha-1)=2cos^2alpha-2$

or ${cos}^{2} θ = \frac{{cos}^{2} α - 1}{cos α - 1} = cos α + 1$ $cos^2theta=(cos^2alpha-1)/(cosalpha-1)=cosalpha+1$

or ${cos}^{2} θ = 2 {cos}^{2} (\frac{α}{2})$ $cos^2theta=2cos^2(alpha/2)$

and $cos θ = \sqrt{2} cos (\frac{α}{2})$ $costheta=sqrt2cos(alpha/2)$

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If $sec (θ + α) + sec (θ - α) = 2 sec (θ)$ $sec(theta+alpha)+sec(theta-alpha)=2sec(theta)$ ,prove that $cos (θ) = \sqrt{2} cos (\frac{α}{2})$ $cos(theta)=sqrt2 cos(alpha/2)$ ?

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Explanation:

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If sec(θ+α)+sec(θ−α)=2sec(θ)sec(theta+alpha)+sec(theta-alpha)=2sec(theta),prove that cos(θ)=√2cos(α2)cos(theta)=sqrt2 cos(alpha/2)?

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Explanation:

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If $sec (θ + α) + sec (θ - α) = 2 sec (θ)$ $sec(theta+alpha)+sec(theta-alpha)=2sec(theta)$ ,prove that $cos (θ) = \sqrt{2} cos (\frac{α}{2})$ $cos(theta)=sqrt2 cos(alpha/2)$ ?