Question

√3tanA=3sinA. then find sin²A-cos²A

Answer 1

tanA/sinA=3/√3
(sinA/cosA)(1/sinA)=√3
1/cosA=√3
cosA=1/√3

sin^2A-cos^2A
1-cos^2A-cos^2A
1-2cos^2A
1-2(1/√3)^2
1-2(1/3)
1-2/3
1/3

Answer 2

$\sin^2 A-\cos^2 A=\dfrac{1}{3}$

Step-by-step explanation:

We have,

$\sqrt{3}\tan A=3\sin A$

To find, $\sin^2 A-\cos^2 A=?$

∴ $\sqrt{3}\dfrac{\sin A}{\cos A} =3\sin A$

[∵ $\tan A=\dfrac{\sin A}{\cos A}$]

⇒ $\dfrac{1}{\cos A} =\sqrt{3}$

⇒ $\cos A=\dfrac{1}{\sqrt{3}}$

∴$\sin^2 A-\cos^2 A$

$=1-\cos^2 A-\cos^2 A$

$=1-2\cos^2 A$

Put $\cos A=\dfrac{1}{\sqrt{3}},$ we get

$=1-2(\dfrac{1}{\sqrt{3}})^2$

$=1-2\times \dfrac{1}{3} =1-\dfrac{2}{3}$

$=\dfrac{3-2}{3}=\dfrac{1}{3}$

Hence, $\sin^2 A-\cos^2 A=\dfrac{1}{3}$.