# If sin A= 1/(sqrt(2)), what is cot A/csc A?

Nov 6, 2014

By the trig identities $\cot \theta = \frac{\cos \theta}{\sin \theta}$ and $\csc \theta = \frac{1}{\sin \theta}$,

$\frac{\cot A}{\csc A} = \frac{\frac{\cos A}{\sin A}}{\frac{1}{\sin A}}$

by multiplying the numerator and the denominator by $\sin A$,

$= \cos A$

by the trig identity ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$,

$= \pm \sqrt{1 - {\sin}^{2} A}$

by $\sin A = \frac{1}{\sqrt{2}}$,

$= \pm \sqrt{1 - {\left(\frac{1}{\sqrt{2}}\right)}^{2}} = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}$

I hope that this was helpful.