Question

tanA+cotA=2cosec2A

Answer 1

$$\tan A + \cot A = 2 \cosec2A$$

Using the definitions of the trigonometric functions

$$\tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\sin A}{\cos A}, \quad \text{so}$$ $$LH = \tan A + \cot A = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 x + \cos^2 x}{\cos A \sin A}$$

Using the Pythagorean identities

$$\sin^2 x + \cos^2 x = 1$$

Using the double-angle formulae

$$\cos A \sin A = \frac{1}{2} \sin 2A$$ $$LH = \frac{\sin^2 x + \cos^2 x}{\cos A \sin A} = \frac{1}{\frac{1}{2} \sin 2A} = \frac{2}{\sin 2A}$$

Using the definitions of the trigonometric functions

$$\cosec2A = \frac{1}{\sin 2A}, \quad \text{so}$$ $$RH = \frac{2}{\sin 2A}, \quad \text{so} \quad LH = RH$$