if sec theta is equals to X + 1 by 4 x prove that sec theta + tan theta is equals to 2 x or 1 by 2 x
Answers
Step-by-step explanation:
Given: secθ = x + (1/4x)
On squaring both sides, we get
⇒ sec²θ = x² + (1/16x²) + (1/2)
∴ sec²θ - tan²θ = 1 (or) sec²θ = 1 + tan²θ
⇒ 1 + tan²θ = x² + (1/16x²) + (1/2)
⇒ tan²θ = x² + (1/16x²) + (1/2) - 1
⇒ tan²θ = x² + (1/16x²) - 1/2
⇒ tan²θ = (x - 1/4x)²
⇒ tanθ = ± (x - 1/4x)
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)
= 1/2x.
Hope it helps!
Given –
secA = x + 1/4x
As 1 + tan^2A = sec^2A
tan^2A = sec^2A – 1
Therfore, tan^2A = (x + 1/4x)^2 – 1
= x^2 + 2*x*1/4x + 1/16x^2 – 1
= x^2 + 1/2 + 1/16x2 – 1
= x^2 + 1/16x^2 – 1/2
= (x – 1/4x)^2
Therefore, tan^2A = x – 1/4x or tan^2A = - (x – 1/4x)
Substitute the value of secA and tanA in the given equation secA + tanA
LHS = secA + tanA
= x + 1/4x + x – 1/4x
= 2x
= RHS
Or
LHS = secA + tanA
=x + 1/4x -x + 1/4x
= 2/4x
= 1/2x
= RHS