Question

Prove that:
`(cos(60^(@)-A)+sin(30^(@)-A))/(cos(30^(@)-A)-sin(60^(@)-A))= cotA`

Answer 1

LHS =`(cos(60^(@)-A)+sin(30^(@)-A))/(cos(30^(@)-A)-sin(60^(@)-A))`
=`(sin{90^(@)-(60^(@)-A)}+sin(30^(@)-A))/(sin{90^(@)-(30^(@)-A)}-sin(60^(@)-A))`
`=(sin(30^(@)+A)+sin(30^(@)-A))/(sin(60^(@)+A)-sin(60^(@)-A))`
`(2sin((30^(@)+A)+(30^(@)-A))/(2).cos((30^(@)+A)-(30^(@)-A))/(2))/(2cos((60^(@)+A)+(60^(2)-A))/(2).sin((60^(@)+A)-(60^(@)-A))/(2))`
`=(sin30^(@)cosA)/(cos60^(@)sin60^(@))=(1/2cosA)/(1/2sinA)=(cosA)/(sinA)`
=cotA= RHS Hence Proved.