Question

Prove the following identities (1-16)

$\left(\mathrm{cosec}x-\sin x\right)\left(\sec x-\cos x\right)\left(\tan x+\cot x\right)=1$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\left(\mathrm{cosec}\mathit{}x-\sin \mathit{}x\right)\left(\sec \mathit{}x-\cos x\right)\left(\tan x+\cot x\right) \\ =\left(\frac{1}{\sin \mathit{}x}-\sin \mathit{}x\right)\left(\frac{1}{\cos x}-\cos x\right)\left(\frac{\sin \mathit{}x}{\cos x}+\frac{\cos x}{\sin \mathit{}x}\right) \\ =\left(\frac{1-{\sin }^{2}x}{\sin \mathit{}x}\right)\left(\frac{1-{\cos }^{2}x}{\cos x}\right)\left(\frac{{\sin }^2\mathit{}x+{\cos }^{2}x}{\cos x\sin \mathit{}x}\right) \\ =\left(\frac{{\cos }^{2}x}{\sin \mathit{}x}\right)\left(\frac{{\sin }^2\mathit{}x}{\cos x}\right)\left(\frac{1}{\cos x\sin \mathit{}x}\right) \\ =1 \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$