Question

If $ 2-{{\cos }^{2}}\theta =3\sin \theta \cos \theta $ , $ \sin \theta \ne cos\theta $ , then find the value of $ \cot \theta $

• A. $ \frac{1}{2} $
• B. $ 0 $
• C. $ -1 $
• D. $ 2 $

Answer 1

Answer: (D)

$ 2-{{\cos }^{2}}\theta =3\sin \theta \cos \theta $
Dividing both sides by $ {{\cos }^{2}}\theta $ ,
we get $ 2{{\sec }^{2}}\theta -1=3\tan \theta $
$ 2(1+{{\tan }^{2}}\theta )-1=3\tan \theta $
$ \Rightarrow $ $ 2+2{{\tan }^{2}}\theta -1=3\tan \theta $
$ \Rightarrow $ $ 2{{\tan }^{2}}\theta -3\tan \theta +1=0 $
$ \Rightarrow $ $ 2{{\tan }^{2}}\theta -2\tan \theta -\tan \theta +1=0 $
$ \Rightarrow $ $ 2\tan \theta (\tan \theta -1)-(\tan \theta -1)=0 $
$ \Rightarrow $ $ (2\tan \theta -1)(\tan \theta -1)=0 $
$ \Rightarrow $ $ \tan \theta =\frac{1}{2} $ and $ 1 $
$ \Rightarrow $ $ \cot \theta =2 $ and $ 1 $