Trigonometric Identities Proof?

Question

cot[ (theta) - 90 ] = (-tan(theta))

90 is in degrees.

Thanks for any help ^^

Answer 1

cot = cos/sin
I will use x for theta
and difference formulas for sin and cos
cos(x-90)/sin(x-90)
cosx *cos90+sinx*sin90
over
sinx*cos90-cosxsin90
cos 90=0
sin90=1
cosx *0+sinx*1
over
sinx*0-cox *1
=sinx/-cosx
=-tanx

Answer 2

cot[ t - 90 ] = -tan(t)

There are identities for tan(a + b), but let's just try and rely only on the sine and cosine addition identities.
Choose the more complex side, which is the left hand side.

LHS = cot (t - 90)

Express this as a quotient of cosine and sine, by the definition of cotangent.

LHS = cos(t - 90) / sin(t - 90)

Now, use the cosine and sine subtraction identities. As a reminder:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
sin(a - b) = sin(a)cos(b) - sin(b)cos(a)

LHS = [ cos(t)cos(90) + sin(t)sin(90) ] / [ sin(t)cos(90) - sin(90)cos(t) ]

cos(90) = 0, sin(90) = 1.

LHS = [ cos(t) [0] + sin(t) [1] ] / [ sin(t) [0] - [1] cos(t) ]
LHS = [sin(t)] / [-cos(t)]
LHS = (-1) sin(t)/cos(t)
LHS = -tan(t) = RHS