# 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.

**Explanation:**

Given, sec θ = x + 1 / 4x

Using the trigonometric identity, 1 + tan^{2 }θ = sec^{2 }θ

==> tan^{2 }θ = sec^{2 }θ - 1

Using sec^{2 }θ = (x + 1/4 x)^{2}, we get

==> 1 + tan^{2 }θ = x^{2} + 1/16 x^{2} + 1/2

tan^{2 }θ = x^{2} + 1/16 x^{2 }- 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.

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