Question

cot A+cosec A-1/cot A-cosec A+1=1+cos A/sin A

Answer 1

Question:

$$\cot A + \csc A - \frac {1}{\cot A} - \csc A + 1 = 1 + \frac {\cos A}{\sin A}$$

Solution:

Take the left side of the expression and simplify it:

$$\begin{array}{l} \cot A + \csc A - \frac {1}{\cot A} - \csc A + 1 = \cot A - \frac {1}{\cot A} + 1 = \frac {\cos A}{\sin A} - \frac {1}{\frac {\cos A}{\sin A}} + 1 = \frac {\cos A}{\sin A} - \frac {\sin A}{\cos A} + 1 \\ = \frac {\cos^ {2} A - \sin^ {2} A}{\sin A \cos A} + 1 \\ \end{array}$$

Substitute this into the initial expression:

$$\begin{array}{l} \frac {\cos^ {2} A - \sin^ {2} A}{\sin A \cos A} + 1 = 1 + \frac {\cos A}{\sin A} \\ \frac {\cos^ {2} A - \sin^ {2} A}{\sin A \cos A} = \frac {\cos A}{\sin A} \\ \frac {\cos^ {2} A - \sin^ {2} A}{\sin A \cos A} - \frac {\cos A}{\sin A} = 0 \\ \frac {\cos^ {2} A - \sin^ {2} A - \cos^ {2} A}{\sin A \cos A} = 0 \\ \frac {\sin A}{\cos A} = 0 \\ \sin A = 0 \\ \end{array}$$

But $\sin A$ can't be equal to zero, because $\cot A$ doesn't exist in this case. So our equation has no roots.