1.

Prove that `(cos7x+cos5x)/(sin7x-sin5x)=cotx`.

Answer» Here, we will use the following formulas:
`sin A - sin B = 2sin((A-B)/2)cos((A+B)/2)`
`cos A + cos B = 2cos((A-B)/2)cos((A+B)/2)`

`L.H.S. = (cos7x+cos5x)/(sin7x-sin5x)`
`=(2cos((7x+5x)/2)cos((7x-5x)/2))/(2cos((7x+5x)/2)sin((7x-5x)/2))`
`=cos(x)/sin(x)=cotx=R.H.S.`


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