If sec θ = x + 1 / 4x, then prove that sec θ + tan θ = 2x or 1 / 2x
We need to consider two different cases.
Answer: The value of sec θ + tan θ = 2x or 1/2x as proved below.
We can make use of the trigonometric identity.
Given, sec θ = x + 1 / 4x
Using the trigonometric identity, 1 + tan2 θ = sec2 θ
==> tan2 θ = sec2 θ - 1
Using sec2 θ = (x + 1/4 x)2, we get
==> 1 + tan2 θ = x2 + 1/16 x2 + 1/2
tan2 θ = x2 + 1/16 x2 - 1/2 = (x - 1/4x)2
Taking square root on both sides, we get tan θ = ± x - 1/4x
(i) if tan θ = (x - 1/4x) and sec θ = (x + 1/4x), we have tan θ + sec θ = 2x ,
(ii) if tan θ = - (x - 1/4x) and sec θ = (x + 1/4x), we have tan θ + sec θ = 1/4x + 1/4x = 1/2x
Hence proved that the value of sec θ + tan θ is 2x or 1/2x.