If sin θ=45, what is the value of cotθ + cosecθ?

Given: $\sin \theta=\frac{4}{5}$

We know that,

$\sin ^{2} \theta+\cos ^{2} \theta=1$

$\Rightarrow\left(\frac{4}{5}\right)^{2}+\cos ^{2} \theta=1$

$\Rightarrow \frac{16}{25}+\cos ^{2} \theta=1$

$\Rightarrow \cos ^{2} \theta=1-\frac{16}{25}$

$\Rightarrow \cos ^{2} \theta=\frac{9}{25}$

$\Rightarrow \cos \theta=\frac{3}{5}$

We have,

$\cot \theta+\operatorname{cosec} \theta=\frac{\cos \theta}{\sin \theta}+\frac{1}{\sin \theta}$

$=\frac{\left(\frac{3}{5}\right)}{\left(\frac{4}{5}\right)}+\frac{1}{\left(\frac{4}{5}\right)}$

$=\frac{3}{4}+\frac{5}{4}$

$=2$

Hence, the value of $\cot \theta+\operatorname{cosec} \theta$ is 2 .