Question

What is $\cot A+\mathrm{cosec}A$ equal to ?

Answer 1

Find equivalent of $\mathbf{cot}\mathbf{}\mathbf{A}\mathbf{+}\mathbf{cosec}\mathbf{}\mathbf{A}$:

$$\begin{gathered} \cot \mathrm{A}+\mathrm{cosec}\mathrm{A}=\frac{\cos \mathrm{A}}{\sin \mathrm{A}}+\frac{1}{\sin \mathrm{A}}\mathbf{}\mathbf{\left[}\begin{array}{c}\mathbf{\because }\mathbf{cot}\mathbf{}\mathbf{A}\mathbf{=}\frac{\mathbf{cos}\mathbf{}\mathbf{A}}{\mathbf{sin}\mathbf{}\mathbf{A}}\mathbf{,}\\ \mathbf{cosec}\mathbf{}\mathbf{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{sin}\mathbf{}\mathbf{A}}\end{array}\mathbf{\right]} \\ =\frac{1+\cos \mathrm{A}}{\sin \mathrm{A}} \\ =\frac{2{\cos }^2\left({\displaystyle \frac{A}{2}}\right)}{2\sin \left(\frac{\mathrm{A}}{2}\right)\cos \left(\frac{A}{2}\right)}\mathbf{}\mathbf{\left[}\begin{array}{c}\mathbf{\because }\mathbf{1}\mathbf{+}\mathbf{cos}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{2}\mathbf{}{\mathbf{cos}}^{\mathbf{}\mathbf{2}}\mathbf{}\left(\frac{A}{2}\right)\mathbf{,}\\ \mathbf{sin}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{}\left(\frac{A}{2}\right)\mathbf{}\mathbf{cos}\mathbf{}\left(\frac{A}{2}\right)\end{array}\mathbf{\right]} \\ =\frac{\cos \left({\displaystyle \frac{A}{2}}\right)}{\sin \left(\frac{\mathrm{A}}{2}\right)} \\ =\cot \left(\frac{A}{2}\right) \end{gathered}$$

Hence, the value of $\mathbf{cot}\mathbf{}\mathbf{A}\mathbf{+}\mathbf{cosec}\mathbf{}\mathbf{A}\mathbf{=}\mathbf{cot}\mathbf{}\left(\frac{A}{2}\right)$