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Question
MathsClass 10

If \( \sqrt{3}tan\theta =3sin\theta ,\) then the value of \( si{n}^{2}\theta -co{s}^{2}\theta \) is
(A) \( \frac{1}{2}\)
(B) \( \frac{1}{3}\)
(C) \( \frac{1}{4}\)
(D) \( \frac{1}{5}\)

Answer

(B)
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Solution

\( \sqrt{3}tan\theta =3sin\theta \)
\( \sqrt{3 }\frac{sin\theta }{cos\theta }=3sin\theta \)
\( cos\theta =\frac{\sqrt{3}}{3}=cos\theta =\frac{1}{\sqrt{3}}\)
\( \therefore\;si{n}^{2}\theta +co{s}^{2}\theta =1\)
\( si{n}^{2}\theta =1-co{s}^{2}\theta \)
\( =si{n}^{2}\theta -co{s}^{2}\theta \)
\( =1-co{s}^{2}\theta -co{s}^{2}\theta \)
\( =1-2\times {\left(\frac{1}{\sqrt{3}}\right)}^{2}\)
\( =1-2\times \frac{1}{3}=1-\frac{2}{3}=\frac{1}{3}\)
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