Question

sinA/1-cos
A+tanA/1+cosA=2secAcosecA+cotA prove the identity

Answer 1

1. $\frac{\sin A}{1 - \cos A} + \frac{\tan A}{1 + \cos A} = 2\sec A\cos ecA + \cot A$ . Prove the identity.

**Solution.**

The left-hand part of the identity:

$$\begin{array}{l} \frac {\sin A}{1 - \cos A} + \frac {\tan A}{1 + \cos A} = \frac {\sin A (1 + \cos A) + \tan A (1 - \cos A)}{(1 - \cos A) (1 + \cos A)} = \frac {\sin A + \sin A \cos A + \tan A - \sin A}{1 - \cos^ {2} A} = \\ = \frac {\sin A \cos A + \tan A}{\sin^ {2} A} = \frac {\sin A \left(\cos A + \frac {1}{\cos A}\right)}{\sin^ {2} A} = \frac {\cos^ {2} A + 1}{\cos A} = \frac {\cos^ {2} A + 1}{\sin A \cos A}. \end{array}$$

The right-hand part of the identity:

$$2 \sec A \cos e c A + \cot A = \frac {2}{\cos A \sin A} + \frac {\cos A}{\sin A} = \frac {2 + \cos^ {2} A}{\sin A \cos A}.$$

**Answer:** the identity is wrong.