Prove that 1cosec A−cotA−1sinA=1sinA−1cosec A+cotA
To prove that 1cosec A−cotA−1sinA=1sinA−1cosec A+cotA
i.e. To prove that 1cosec A−cotA+1cosec A+cotA=1sinA+1sinA
i.e. To prove that cosec A+cotA+cosec A−cotA(cosec A+cotA)(cosec A−cotA)=2sinA
i.e. To prove that 2cosec Acosec2A−cot2A=2cosec A
Now, 1+cot2A=cosec2A
⟹ cosec2A−cot2A=1
∴ To prove that \; 2 \; \text{cosec }A = 2 \; \text{cosec A} \;\; which is true.
\therefore \; \dfrac{1}{\text{cosec }A - \cot A} - \dfrac{1}{\sin A} = \dfrac{1}{\sin A} - \dfrac{1}{\text{cosec }A + \cot A}
Hence proved.