Question

In a ΔАBC, if ∠B = 90° and tan A = 1 then prove that 2 sin A cos A = 1.

Answer 1

Given: tan A = 1

To prove: 2 sin A cos A = 1

Proof:

Given, а ΔАВC in which tan A = 1.

Then, tan A = 1 ⇒ `(BC)/(AB) = 1`.

Let BC = x. Then, AB = x.

∴ AC² = (AB² + BC²)

= (x² + x²)

= 2x²

⇒ `AC = sqrt(2x^2)`

⇒ `AC = sqrt(2)x`.

∴ `2 sin A cos A = 2 xx ((BC)/(AC) xx (AB)/(AC))`

= `(2 xx x/(sqrt(2)x) xx x/(sqrt(2)x))`

= 1