Question

If $\sin \theta +\cos \theta =\sqrt{3}$, then find the value of $\tan \theta +\cot \theta$.

• A. $0$
• B. $1$
• C. $2$
• D. $3$
• E. $Noneofthese$

Answer 1

Answer: (B)

$1$

$\begin{array}{c}\sin \theta +\cos \theta =\sqrt{3}\\ squaringonbothside\\ \Rightarrow {\left(\sin \theta +\cos \theta \right)}^2={\sqrt{3}}^2\\ \Rightarrow \left({\sin }^2\theta +{\cos }^2\theta \right)+2\sin \theta .\cos \theta =3\\ \Rightarrow 2\sin \theta .\cos \theta =2\left[{\sin }^2\theta +{\cos }^2\theta =1\right]\\ \therefore \sin \theta .\cos \theta =1\\ Then,\tan \theta +\cot \theta \\ \tan \theta +\cot =\frac{{\sin }^2\theta +{\cos }^2\theta }{\left(\sin \theta .\cos \theta \right)}\\ \frac{1}{1}=1\end{array}$