Question

Sec theta - tan theta= x show that sin theta =1/2[x+1/x] and tan theta= 1/2[1/x-x]

Answer 1

Modified question should be

If secθ - tanθ = x, then find sinθ, cosθ and tanθ.

Formula :

• sec²θ - tan²θ = 1

• a² - b² = (a + b) (a - b)

• sin²θ + cos²θ = 1

• (a + b)² - 4ab = (a - b)²

Solution :

Given that, secθ - tanθ = x ...(i)

We know that, sec²θ - tan²θ = 1

or, (secθ + tanθ) (secθ - tanθ) = 1

or, (secθ + tanθ) * x = 1

or, secθ + tanθ = 1/x ...(ii)

Now, (i) + (ii) gives

2 secθ = x + 1/x

or, secθ = (x² + 1)/(2x)

or, cosθ = 2x/(x² + 1)

Then, sin²θ = 1 - cos²θ

= 1 - {2x/(x² + 1)}²

= {(x² + 1)² - 4x²} / (x² + 1)²

= (x² - 1)² / (x² + 1)²

= {(x² - 1) / (x² + 1)}²

Then, sinθ = (x² - 1) / (x² + 1)

Now, tanθ = sinθ / cosθ

= {(x² - 1) / (x² + 1)} / {2x / (x² + 1)}

= (x² - 1)/(2x)

Hence, proved.

Answer 2

Hope it helps

Thanks for asking