Question

in a right angle triangle ABC right angled at B if tan A=1 then show that 2 sinA cos A=1​

Answer 1

Refer to the attachment:

Given: In ∆ABC , <B = 90°

and, tanA = 1

To prove: 2sinA cosA = 1

Proof: tanA = 1

=> $\rm\frac{BC}{AB} = 1$

=> BC = AB ---> (a)

By Pythagoras theorem:

AC² = AB² + BC²

=> AC² = AB² + (AB)² {From (a)}

=> AC² = 2AB²

=> AC = $\rm\sqrt{2AB^2}$

=> AC = $\rm\sqrt{2}$AB

2sinA cosA = 2 × $\frac{BC}{AC} ×\frac{AB}{AC}$

= 2 × $\rm\frac{AB}{\sqrt{2}AB} ×\frac{AB}{\sqrt{2}AB}$

= 2 × $\rm\frac{1}{\sqrt{2}} ×\frac{1}{\sqrt{2}}$

= 1

LHS = RHS

Hence proved