Prove the identities — task. Maths CBSE Live product, Class 10 (2026-27).

If \(\cot \theta + \tan \theta = x\) and \(\sec \theta - \cos \theta = y\), then verify that \((x^2 y)^{\frac{2}{3}} - (x y^2)^{\frac{2}{3}} = 1\).

Proof:

\begin{array}{l} x = \cot θ + \tan θ \\ = \frac{i θ}{i θ} + \frac{i θ}{i θ} \\ x^{2} = \frac{i}{\sin^{2} θ . i^{2} θ} \\ y = \sec θ - \cos θ \\ = \frac{1}{i θ} - \cos θ \\ = \frac{i^{2} θ}{i θ} - - - - - - (2) \\ x^{2} y = \frac{i}{i^{3} θ} \\ {(x^{2} y)}^{\frac{2}{3}} = \frac{i}{i^{2} θ} \\ x y^{2} = \frac{i^{3} θ}{i^{3} θ} \\ {(x y^{2})}^{\frac{2}{3}} = \frac{i^{2} θ}{i^{2} θ} \\ {(x^{2} y)}^{\frac{2}{3}} - {(x y^{2})}^{\frac{2}{3}} = i \end{array}

\(=RHS\)

Hence, proved.

Did you find an error?

5. Prove the identities