Question

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

Answer 1

Prove that the following identities:

$\frac{1+\cos ecA}{cotA}=\frac{\cot A-1-\cos ecA}{\cos ecA+1+\cot A}$

LHS=RHS

Solve the RHS side

Using a given formula-

$\cos e{c}^{2}A-{\cot }^{2}A=1$

$\frac{\cot A-1-\cos ecA}{\cos ecA+1+\cot A}=\frac{\left[\right(\cot A-\cos ecA)-(\cos e{c}^{2}A-{\cot }^{2}A\left)\right]}{\cos ecA+1+\cot A}$

$=\frac{\left[\right(\cot A-\cos ecA)+(\cot A-\cos ecA\left)\right(\cot A+\cos ecA\left)\right]}{\cos ecA+1+\cot A}$

$=\frac{(\cot A-\cos ecA)(1+\cot A+\cos ecA)}{(\cos ecA+1+\cot A)}$

$=(\cot A-\cos ecA)$

$=\frac{\cos A}{\sin A}-\frac{1}{\sin A}$

$=\frac{\cos A-1}{\sin A}$

Hence, proved.