Question

9. If 7 sin2θ+ cos2θ=44, show that tan=1/√3

Answer 1

${\bold{\underline{\underline{Solution:}}}}$

Given :

$7\sin^{2}\theta+3\cos^{2}\theta=4$

$\implies7\sin^{2}\theta+3\cos^{2}\theta=4\\\\ \implies7(1-cos^{2}\theta)+3\:cos^{2} \theta=4\\\\ \implies7-7\:cos^{2}\theta+3\:cos^{2} \theta=4\\\\ \implies7-4\:cos^{2}\theta=4\\\\ \implies4\:cos^{2}\theta=3\\\\ \implies\cos^{2}\theta =\frac{3}{4}\:\:.....(1)\\\\ \implies1-sin^{2}\theta=\frac{3}{4}\\\\\implies\:sin^{2}\theta=1-\frac{3}{4}=\frac{1}{4}\:\:\:.....(2)$

We Know that :

$\bold{\boxed{{\tan}^{2}\theta =</p><p>\frac{{\sin}^{2}\theta}{{\cos}^{2}\theta}}}$

$\implies$ $\sqrt{\frac{\frac{1}{4}}{\frac{3}{4}}}$

$\implies\sqrt{\frac{1}{3}}=\frac{1}{\sqrt{3}}</p><p>$

Hence Proved.

Answer 2

Answer:

{\bold{\underline{\underline{Solution:}}}}

Given :

7\sin^{2}\theta+3\cos^{2}\theta=4

\implies7\sin^{2}\theta+3\cos^{2}\theta=4\\\\ \implies7(1-cos^{2}\theta)+3\:cos^{2} \theta=4\\\\  \implies7-7\:cos^{2}\theta+3\:cos^{2} \theta=4\\\\ \implies7-4\:cos^{2}\theta=4\\\\ \implies4\:cos^{2}\theta=3\\\\ \implies\cos^{2}\theta =\frac{3}{4}\:\:.....(1)\\\\ \implies1-sin^{2}\theta=\frac{3}{4}\\\\\implies\:sin^{2}\theta=1-\frac{3}{4}=\frac{1}{4}\:\:\:.....(2)

We Know that :

\bold{\boxed{{\tan}^{2}\theta =</p><p>\frac{{\sin}^{2}\theta}{{\cos}^{2}\theta}}}

\implies \sqrt{\frac{\frac{1}{4}}{\frac{3}{4}}}

\implies\sqrt{\frac{1}{3}}=\frac{1}{\sqrt{3}}</p><p>

Hence Proved

Step-by-step explanation: