prove identity is true and please show steps?

Question

sin t csc t (Pi/2-t) = tan t

Answer 1

I think you've written your problem incorrectly. This, in all likelihood, should be:

sin(t)csc(Pi/2-t) = tan(t)

THAT has a solution:

We know that tan(t) = sin(t)/cos(t), so the "sin(t)"'s cancel ==>

csc(Pi/2-t) = 1/cos(t)

Now, we know that Cos(t) = Sin(Pi/2 - t) {Edited to add: because we know that the circular functions sin and cos are Pi/2 Radians out of phase with one another} AND we know that Csc(x) = 1/sin(x)

Let x=Pi/2-t ==> csc(x) = 1/sin(x), which is true.

Answer 2

csct si =1/sin so sin(pi/2-t) = sinpi/2*cost -cos pi/2 sin t
but sinpi/2=1 and cospi/2=0.
The denominator becomes cost and sint/cost=tan t