Question

show that 1-sin60°/cos60°=tan60°-1/tan60°+1

Answer 1

LHS = (1 -sin60°)/cos60°
= (1 - √3/2)/(1/2) = (2-√3)

RHS = ( tan60° -1)/(tan60° +1)

=( √3 -1)/(√3 +1)

=(√3 -1)(√3 -1)/(√3 +1)(√3 -1)

=( √3² +1² -2√3)/(√3²-1)

=( 4-2√3)/(2)

= 2 -√3

LHS = RHS
hence proved

Answer 2

sin 60 = √3/2
cos 60 = 1/2
tan 60 = √3

lhs = (1-sin 60) /cos 60

= (1-√3/2) /(1/2) 
= 2-√3

rhs = (tan60 -1) /(tan60 +1)

= (√3-1)/(√3 +1)
= (√3-1) (√3 -1)/[(√3+1)(√3-1)]
=(√3 -1)²/[(√3)² -1²]

= (3+1 -2√3) /(3-1)
=(4 -2√3)/2
=2-√3
therefore 
lhs = rhs