Question

If $\csc \theta = 2$, then find the value of $\cot \theta + \dfrac{\sin \theta}{1 + \cos \theta}$.

Hint:

When $\csc \theta$ or $\sec \theta$ is given, convert it to $\sin \theta$ or $\cos \theta$ and then use identities to simplify.

Answer 1

Step 1: Given.
$\csc \theta = 2 \Rightarrow \sin \theta = \dfrac{1}{2}$.
Step 2: Find $\cos \theta$.
\[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \left(\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \Rightarrow \frac{1}{4} + \cos^2 \theta = 1 \Rightarrow \cos^2 \theta = \frac{3}{4} \Rightarrow \cos \theta = \frac{\sqrt{3}}{2} \] Step 3: Find $\cot \theta$.
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
Step 4: Evaluate the expression.
\[ \cot \theta + \frac{\sin \theta}{1 + \cos \theta} = \sqrt{3} + \frac{\frac{1}{2}}{1 + \frac{\sqrt{3}}{2}} = \sqrt{3} + \frac{1}{2} \times \frac{1}{\frac{2 + \sqrt{3}}{2}} = \sqrt{3} + \frac{1}{2 + \sqrt{3}} \] Rationalize: \[ \frac{1}{2 + \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = 2 - \sqrt{3} \] Hence, \[ \sqrt{3} + (2 - \sqrt{3}) = 2 \]
Step 5: Conclusion.
The value of the given expression is $2$.