Question

If sec theta = x+1/4x, find sec theta + tan theta​

Answer 1

Solution :-

sec θ = x + 4/4x

We know that

tan² θ = sec² θ - 1

Substituting the value of sec θ

⇒ tan² θ = (x + 1/4x)² - 1

It can be written as

⇒ tan²θ = (x + 1/4x)² - 4(x)(1/4x)

⇒ tan²θ = (x - 1/4x)²

[ Because (a + b)² - 4ab = (a - b)² ]

Taking square root on both sides

⇒ √tan²θ = ± √(x - 1/4x)²

⇒ tanθ = ± (x - 1/4x)

⇒ tanθ = (x - 1/4x) or - (x - 1/4x)

⇒ tanθ = (x - 1/4x) or (- x + 1/4x)

Now, secθ + tanθ

When tanθ = x - 1/4x

secθ + tanθ = x + 1/4x + x - 1/4x = 2x

When tanθ = - x + 1/4x

secθ + tanθ = x + 1/4x - x + 1/4x = 2/4x = 1/2x

Therefore the value of secθ + tanθ is 2x or 1/2x.