To prove a statement is not an identity find a value that ma

To prove a statement is not an identity, find a value that makes the statement false. Which angle proves csc^2 x - cot x = 1 is not an identity. Pie/2 pie/4 7 pie/4 3 pie/2

Solution

csc^2x - cotx=1

1/sin^2x - cosx/sinx =1

for x= 7pi/4 sin(7pi/4) = sin(2pi -pi/4) = -sinpi/4 = -1/sqrt2

cos(2pi-pi/4) = cospi/4

cot(pi/4) = cospi/4/-sinpi/4 = -1

So, csc^2x - cotx = 2- (-1) = 3

So, for x= 7pi/4 does not hold