Question #e026e
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"What is Arithmetic Mean?"
We need
#cscx=1/sinx#
#cotx=cosx/sinx#
#cos^2x+sin^2x=1#
Therefore
#LHS =(1/(cscx-cotx))-(1/(cscx+cotx))#
#=(1/(1/sinx-cosx/sinx))-(1/(1/sinx+cosx/sinx))#
#=(sinx/(1-cosx))-(sinx/(1+cosx))#
#=sinx((1+cosx-1+cosx))/(1-cos^2x)#
#=sinx/sin^2x*2cosx#
#=2/(sinx/cosx)#
#=2/tanx#
#=RHS#
#QED#
#(1/(cscx - cotx)) -(1/(cscx + cotx)) = 2/tanx#
Multiply the fractions on the left side by the conjugates of the denominators (which is the same as multiplying by one):
#"Left side:"#
#frac{1}{cscx-cotx} color(blue)(*frac{cscx+cotx}{cscx+cotx}) - frac{1}{cscx + cotx} color(blue)(*frac{cscx-cotx}{cscx-cotx})#
# "LS"= frac{(cscx+cotx) - (cscx - cotx)}{csc^2 x -cot^2 x}#
Using the pythagorean trig identity, we know that #csc^2x - cot^2x = 1#
#"LS" = frac{color(red)(cscx) + cotx color(red)(-cscx) + cotx}{color(blue)((csc^2x - cot^2x))}#
#"LS" = 2cotx#
#"LS" = 2/tanx#
#="Right side"#
#QED#
