Prove this Trig identity?

The answer is as follows:

sin(A - B) = sinAcosB - cosAsinB

so, sin(π/2 - x) = sin(π/2)cosx - cos(π/2)sinx

As sin(π/2) = 1 and cos(π/2) = 0 we have:

sin(π/2 - x) = (1)cosx - (0)sinx => cosx

Then, sin(x)sin(π/2 - x) + cos²(x)cotx becomes:

sin(x)cos(x) + cos²(x)cotx

Now, cotx = cosx/sinx so,

sin(x)cos(x) + cos²(x)cosx/sinx

=> [sin²(x)cos(x) + cos³x]/sinx

i.e. cosx[sin²(x) + cos²x]/sinx

Now, sin²(x) + cos²x = 1 so we reduce to, 

cosx/sinx = cotx => left side

:)>

Notation : RTP stands for ''required to prove''.

RTP: cot(x) = sin(x)sin(pi/2 -x) + cos^2(x)cot(x), ie., for tan(x) =/= 0;
RTP: tan(x)cot(x) = tan(x)sin(x)sin(pi/2 -x) + cos^2(x)cot(x)tan(x), ie.,;
RTP: 1 =(sin(x)/cos(x))sin(x)sin(pi/2 -x) +cos^2(x). Note: sin(pi/2 -x) = cos(x).
RTP: 1 = (sin^2(x)/cos(x))*cos(x) + cos^2(x), ie.,;
RTP: 1 = sin^2(x) + cos^2(x), a standard identity.

I hope your teacher has taught you the trigonometric functions sine, cosine,
tangent, cotangent , secant & cosecant. Functions in the following pairs are
reciprocals of each other : (sin,csc),(cos,sec),(tan,cot). Suppose A & B are 2
different angles. Are the following relationships familiar to you?
sin(A+B) = sinAcosB + cosAsinB;
sin(A- B) = sinAcosB -  cosAsinB;
cos(A+B)= cosAcosB - sinAsinB;
cos(A- B)= cosAcosB +sinAsinB;
tan(A+B) = (tanA + tanB)/(1- tanA*tanB);
tan(A- B) = (tanA -  tanB)/(1+tanA*tanB); 
Good luck in finding a great teacher. I'll relate an experience I had one time.
I was asked to volunteer as a high school math teacher. I spent a week 
teaching the grade 12 class how to solve their math problems. They were thrilled to have someone great at explaining the ''how to'' involved in problem
solving. Their former time with an English teacher trying to teach them math
was most frustrating, they said. I went to the principal and asked if there were any chance of me getting paid for my work. She asked me if I had a teaching
certificate and I said no. She said she could only pay those with a teaching
certificate to teach. My honors math diploma was irrelevant. They wanted to
have me teaching but didn't want to pay for it. Consequently, I never returned to the high school. I felt sorry for the students but was unwilling to be treated
like some fool.