If \(\frac{3}{\sqrt {3}}\)cotθ=1, then find the value of \(\frac{2-sin^2θ}{1-cos^2θ}\)+(cosec²θ-secθ)

1

\(\frac{3}{\sqrt {3}}\)cotθ=1

\(\frac{\sqrt {3}×\sqrt {3}}{\sqrt {3}}\)cotθ=1

cotθ=\(\frac{1}{\sqrt {3}}\)=60°

⇒ \(\frac{2-sin^2θ}{1-cos^2θ}\)+(cosec²θ-secθ) = \(\frac{2-(\frac{\sqrt {3}}{2})^2}{1-(\frac{1}{2})^2}\)+(\(\frac{2}{\sqrt {3}}\))²-2

⇒ \(\frac{2-\frac{3}{4}}{1-\frac{1}{4}}\)+\(\frac{4}{3}\)-\(\frac{2}{3}\)

⇒ \(\frac{\frac{5}{4}}{\frac{3}{4}}\)+[\(\frac{4}{3}\)-\(\frac{2}{3}\)]

= 1