Question

The value of (cosec θ – sin θ) (secθ – cos θ) (tan θ + cot θ) is _________.

Answer 1

$$\begin{gathered} \left(\mathrm{cosec}\theta -\sin \theta \right)\left(\sec \theta -\cos \theta \right)\left(\tan \theta +\cot \theta \right) \\ =\left(\frac{1}{\sin \theta }-\sin \theta \right)\left(\frac{1}{\cos \theta }-\cos \theta \right)\left(\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }\right) \\ =\left(\frac{1-{\sin }^2\theta }{\sin \theta }\right)\left(\frac{1-{\cos }^2\theta }{\cos \theta }\right)\left(\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }\right) \end{gathered}$$
$$\begin{gathered} =\frac{{\cos }^2\theta }{\sin \theta }\times \frac{{\sin }^2\theta }{\cos \theta }\times \frac{1}{\sin \theta \cos \theta }\left({\cos }^2\theta +{\sin }^2\theta =1\right) \\ =1 \end{gathered}$$

The value of (cosec θ – sin θ) (secθ – cos θ) (tan θ + cot θ) is ___1___.