Question

Prove that

$\left(\cos ecA-\sin A\right)\left(\sec A-\cos A\right)\left(\tan A+\cot A\right)=1$

Answer 1

$\left(\text{cosec}A-\sin A\right)\left(\sec A-\cos A\right)\left(\tan A+\cot A\right)$

$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)$

$=\frac{\left(1-{\sin }^{2}A\right)}{\sin A}\times \frac{\left(1-{\cos }^{2}A\right)}{\cos A}\times \frac{{\sin }^{2}A+{\cos }^{2}A}{\cos A\sin A}$

$=\frac{\left(co{s}^{2}A\right)}{\sin A}\times \frac{\left(si{n}^{2}A\right)}{\cos A}\times \frac{1}{\cos A\sin A}=1$

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