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Trigonometry

Prove that 1cosec AcotA1sinA=1sinA1cosec A+cotA


To prove that 1cosec AcotA1sinA=1sinA1cosec A+cotA

i.e. To prove that 1cosec AcotA+1cosec A+cotA=1sinA+1sinA

i.e. To prove that cosec A+cotA+cosec AcotA(cosec A+cotA)(cosec AcotA)=2sinA

i.e. To prove that 2cosec Acosec2Acot2A=2cosec A

Now, 1+cot2A=cosec2A

cosec2Acot2A=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.