If $ \sqrt{3} \tan \theta=3 \sin \theta $, find the value of $ \sin ^{2} \theta-\cos ^{2} \theta $
Given:
\( \sqrt{3} \tan \theta=3 \sin \theta \)
To do:
We have to find the value of \( \sin ^{2} \theta-\cos ^{2} \theta \).
Solution:
We know that,
$\tan \theta=\frac{\sin \theta}{\cos \theta}$
This implies,
$\sqrt{3} \tan \theta=3 \sin \theta$
$\sqrt{3} \frac{\sin \theta}{\cos \theta}=3 \sin \theta$
$\sqrt{3}=3 \cos \theta$
$\cos \theta=\frac{\sqrt3}{3}$
$\cos \theta=\frac{1}{\sqrt3}$
$\Rightarrow \sin^{2} \theta=1-\cos^2 \theta$
$=1-(\frac{1}{\sqrt3})^2$
$=1-\frac{1}{3}$
$=\frac{3-1}{3}$
$=\frac{2}{3}$
Therefore,
$\sin ^{2} \theta-\cos ^{2} \theta=\frac{2}{3}-(\frac{1}{\sqrt3})^2$
$=\frac{2}{3}-\frac{1}{3}$
$=\frac{2-1}{3}$
$=\frac{1}{3}$
The value of \( \sin ^{2} \theta-\cos ^{2} \theta \) is $\frac{1}{3}$.
Related Articles
- If \( 3 \cos \theta-4 \sin \theta=2 \cos \theta+\sin \theta \), find \( \tan \theta \).
- Prove that: $\frac{sin \theta-2 sin ^{3} \theta}{2 cos ^{3} \theta-cos \theta}= tan \theta$
- If \( \sqrt{3} \tan \theta=1 \), then find the value of \( \sin ^{2} \theta-\cos ^{2} \theta \).
- If \( \cos \theta+\cos ^{2} \theta=1 \), prove that\( \sin ^{12} \theta+3 \sin ^{10} \theta+3 \sin ^{8} \theta+\sin ^{6} \theta+2 \sin ^{4} \theta+2 \sin ^{2} \theta-2=1 \)
- If \( \sqrt{3} \tan \theta-1=0 \), find the value of \( \sin ^{2}-\cos ^{2} \theta \).
- If \( \tan \theta=\frac{12}{13} \), find the value of \( \frac{2 \sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta} \)
- If \( 3 \cos \theta=1 \), find the value of \( \frac{6 \sin ^{2} \theta+\tan ^{2} \theta}{4 \cos \theta} \)
- If \( \sin \theta=\frac{12}{13} \), find the value of \( \frac{\sin ^{2} \theta-\cos ^{2} \theta}{2 \sin \theta \cos \theta} \times \frac{1}{\tan ^{2} \theta} \)
- If \( \cos \theta=\frac{5}{13} \), find the value of \( \frac{\sin ^{2} \theta-\cos ^{2} \theta}{2 \sin \theta \cos \theta} \times \frac{1}{\tan ^{2} \theta} \)
- If \( \cos \theta=\frac{3}{5} \), find the value of \( \frac{\sin \theta-\frac{1}{\tan \theta}}{2 \tan \theta} \)
- Prove: $\sin ^{6} \theta+\cos ^{6} \theta=1-3 \sin ^{2} \theta \cos ^{2} \theta$.
- If $sin\theta +cos\theta=\sqrt{3}$, then prove that $tan\theta+cot\theta=1$.
- If \( \sin \theta+2 \cos \theta=1 \) prove that \( 2 \sin \theta-\cos \theta=2 . \)
- If \( a \cos ^{3} \theta+3 a \cos \theta \sin ^{2} \theta=m, a \sin ^{3} \theta+3 a \cos ^{2} \theta \sin \theta=n \), prove that \( (m+n)^{2 / 3}+(m-n)^{2 / 3}=2 a^{2 / 3} \)
- If \( \cot \theta=\frac{1}{\sqrt{3}} \), find the value of \( \frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta} \)
Kickstart Your Career
Get certified by completing the course
Get Started