Question

If `sin θ = a/b` then prove that `(sec θ + tan θ) = sqrt((b + a)/(b - a)`.

Answer 1

Given: `sin θ = a/b`

To Prove: `(sec θ + tan θ) = sqrt((b + a)/(b - a))`

Proof [Step-wise]:

1. Put p = sec θ + tan θ.

2. From the identity we have `sin θ = (p^2 - 1)/(p^2 + 1)`.

3. Substitute the given value: `(p^2 - 1)/(p^2 + 1) = a/b`.

4. Cross-multiply: b(p² – 1) = a(p² + 1).

5. Expand and collect p² terms: bp² – b = ap² + a 

⇒ (b – a)p² = a + b

6. Solve for p²: `p^2 = (a + b)/(b − a)`.

7. Hence, `p = ± sqrt((a + b)/(b - a))`. For the usual choice of principal value when sec θ + tan θ is positive, e.g. appropriate quadrant or b > a > 0, take the positive root: `p = sqrt((b + a)/(b − a))`.

Therefore, `(sec θ + tan θ) = sqrt((b + a)/(b − a))` with the sign chosen according to the quadrant/conditions so that sec θ + tan θ is positive.