If $$\cosec \theta = 1.25,$$ then $$\frac{4 \tan \theta - 5 \cos \theta + 1}{\sec \theta + 4 \cot \theta -1} = ?$$

Given that $$\cosec \theta = 1.25  = \dfrac {5}{4}$$

so $$\sin \theta = \dfrac{4}{5} $$

then $$\cos \theta = \dfrac {3}{5} $$, $$\tan \theta = \dfrac {4}{3}$$, $$\cot \theta = \dfrac{3}{4}$$

Hence $$ \dfrac {4 \tan \theta - 5 \cos \theta + 1}{ \sec \theta + 4 \cot \theta -1}$$

$$\Rightarrow \dfrac {4\times \dfrac{4}{3} - 5 \times \dfrac{3}{5} +1} {\dfrac{5}{3} + 4\times \dfrac{3}{4} -1}$$

$$\Rightarrow \dfrac{ \dfrac{16}{3} -2} {\dfrac {5} {3} +2} $$

$$\Rightarrow \dfrac{16-6}{5+6}$$

$$\Rightarrow \dfrac{10}{11} $$ Ans