1/cosec@-cot@ - 1/sin@=1/sin@ - 1/cosec@+cot@ prove it
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MayankDubey1:
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Hi!
We will start from the left-hand side (LHS) and show that you can get to the RHS:
LHS=1/(cscA-cotA)-1/sinA(Given)
=(sinA-(cscA-cotA))/(sinA(cscA-cotA)) (Add the fractions)
=(sinA-1/sinA+cosA/sinA)/(sinA(1/sinA-cosA/sinA))
=((sin^2A-1+cosA)/sinA)/(1-cosA)
=((cosA-cos^2A)/sinA)/(1-cosA) (Use the pythagorean identity)
=((cosA(1-cosA))/sinA)/(1-cosA)
=cosA/sinA
=((cosA(1+cosA))/sinA)/(1+cosA)
=((cosA+cos^2A)/sinA)/(1+cosA) Now use the pythagorean identity
=((cosA+1-sin^2A)/sinA)/(1+cosA)
=(1/sinA+cosA/sinA-sinA)/(1+cosA)
`=(cscA+cotA-sinA)/(sinA(1/sinA+cosA/sinA))`
`=(cscA+cotA-sinA)/(sinA(cscA+cotA))`
`=1/sinA-1/(cscA+cotA)`
=RHS as required.
Hope it helps
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