Question

If tan A=2/3,prove that sin²A+cos²A=1

Answer 1

Solution:-

Given;

$\rm \implies \tan A=\dfrac{2}{3}$

To prove

$\rm \implies \sin^{2} A+\cos^{2} A=1$

Now take

$\rm \implies \tan A=\dfrac{2}{3}=\dfrac{p}{b}$

So

$\rm p= 2 , b=3\: and\: h= x$

Using pythagoras theorem

$\rm \implies h^{2} = p^{2} +b^{2} \\ \implies h^{2}=2^{2} + 3^{2}$

$\rm \implies h=\sqrt{13}$

$\rm p= 2 , b=3\: and\: h= \sqrt{13}$

Now

$\rm \implies \sin A = \dfrac{2}{\sqrt{13} } =\dfrac{p}{h} \\ \rm \implies \cos A = \dfrac{3}{\sqrt{13} }=\dfrac{b}{h}$

put the value on given equation

$\rm \implies \sin^{2} A+\cos^{2} A$

$\rm \implies \bigg(\dfrac{2}{\sqrt{13} } \bigg)^{2} +\bigg(\dfrac{3}{\sqrt{13} } \bigg)^{2} \\ \rm \implies \dfrac{4}{13} +\dfrac{9}{13}$

$\implies\dfrac{13}{13} =1$

Hence proved