# How do you verify sec(x+y)= (secx secy)/(1-tanxtany)?

Jul 20, 2015

Verify $\frac{\sec x . \sec y}{1 - \tan x . \tan y} = \sec \left(x + y\right)$

#### Explanation:

((1/cos x)(1/cos y))/((1 - (sin x.sin y)/(cos x.cos y)) =

= $\frac{1}{\cos x . \cos y - \sin x . \sin y} = \frac{1}{\cos} \left(x + y\right) =$ $\sec \left(x + y\right)$

Jul 20, 2015

Let us start with, x = ${30}^{0}$ = y.
we shall show that both left hand side and right hand side equal in value to 2

#### Explanation:

L.H.S. = sec ( ${30}^{0}$ + ${30}^{0}$) = $\sec {60}^{0}$ = 2 .. (1)
R.H.S. = sec${30}^{0}$sec${30}^{0}$ / [1 - $\tan {30}^{0} \tan {30}^{0}$]
= (2 / $\sqrt{3}$)(2 / $\sqrt{3}$) / [ 1 - (1/$\sqrt{3}$ )(1/$\sqrt{3}$ )]
= (4/3) / {1 - 1/3}
= (4/3)x(3/2)
= 2 ....(2)
From (1) and (2) LHS = RHS.
Hence the verification is complete.