Question

Prove that sin^2A+cos^2A=1

Answer 1

let PERPENDICULAR be x and base be y , So

Hypotenuse = √x² + y² { by Pythagoras theorem }

Now,

we know,

$</p><p>\rightarrow \quad \sin A = \frac{\mathtt{Perpendicular}}{\mathtt{Hypotenuse}}</p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \sin A = \frac{x}{\sqrt{x^{2} + y^{2} }} </p><p></p><p>\\ \\</p><p></p><p>\mathtt{Square \: both \: sides}</p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \sin^{2} = \frac{x^{2}}{x^{2} + y^{2}} \qquad \cdots (i)</p><p></p><p>$

Also,

$</p><p></p><p>\rightarrow \quad \cos A = \frac{\mathtt{Base}}{\mathtt{Hypotenuse}}</p><p></p><p>\\ \\ </p><p></p><p>\rightarrow \quad \cos A = \frac{y}{\sqrt{x^{2} + y^{2} }} </p><p></p><p>\\ \\</p><p></p><p>\mathtt{Square \: both \: sides}</p><p></p><p>\\ \\</p><p></p><p>\rightarrow \quad \cos^{2} = \frac{y^{2}}{x^{2} + y^{2}} \qquad \cdots (ii)</p><p></p><p>$

Adding (i) and (ii)

$</p><p>\Rightarrow \quad \sin^{2} A + \cos^{2} A = \frac{x^{2}}{x^{2} + y^{2}} + \frac{y^{2} }{ x^{2} + y^{2} }</p><p></p><p>\\ \\</p><p></p><p>\Rightarrow \quad \sin^{2} A + \cos^{2} A = \frac{\cancel{x^{2} + y^{2}} }{\cancel{x^{2} + y^{2}}}</p><p></p><p>\\ \\</p><p></p><p>\Rightarrow \quad \sin^{2} A + \cos^{2} A = 1 </p><p></p><p>\\ \\</p><p></p><p>$

Hence, Proved.