Help with Trig

tan(x+450)=tan(x+450-360)=tan(x+90)

tan(x)=sin(x)/cos(x)
@ x=0
at tan(90)
sin(90)=1 and cos(90)=0, so tan(90) is division by zero and it is undefined (it'll happen again at x=180 where x+90=270)

I'm making a stab at a trig identity---an identity is continuous, that is, the curve formed iis smooth and contains no undefined points. Sin and cos are such curves that vary between -1 and 1 for all x. However, tan, cot, sec, and csc all vary between +/- infinity with undefined points at some specific x's. This is the result of trig definitions when sin(x) or cos(x) are equal to zero and you divide (remember that division by zero is undefined)

So it is my guess that an identity is continuous for all x--sin and cos are identities. Sec, csc, tan, and cot are not continuous, so they are not identities.

Rap

Identity: An equation that is satisfied by any number that replaces the letter for which the equation is defined.

Wolfram gives:
tan^2(x) + 1 = sec^2(x)
as an identity.

Continuity is not required. It essentially means that one side of the equation can be substituted for the other in all cases.

sin^2(x) + cos^2(x) = 1 is an identity
but
sin(x) + cos(x) = sqrt(2) is not even though it is true for x=45 degrees.

Tangent of X+90
Raprap hinted at this, but the reason you can't use the tangent angle summation formula is that one of the angles is 90 degrees.

tan(x + a)= (tan x + tan a) / (1 - (tan x)(tan a))

You can't use this formula if a is 90 degrees. On the other hand, just looking at a circle shows you that you should be able come up with a formula for this. You can say that

sin(x+90) = sin(x)cos(90) + sin(90)cos(x) = cos(x)
cos(x+90) = cos(x)cos(90) - sin(x)sin(90) = - sin(x)

so...

tan(x+90) = sin(x+90)/cos(x+90) = -cos(x)/sin(x) = -1/tan(x)

which is -cot(x). Wow thats confusing. Imposible one way but not the other.

Naw it isn't. cot(x)=cos(x)/sin(x) & tan(x)=sin(x)/cos(x) as cos(x) goes to zero sin(x) goes to +/-1 cot(x) goes to 0 and tan(x) goes to +/- infinity.

when sin(x) goes to zero the opposite happens.

Sorry about the identity redherring. Shoulda consulted Wolfram, Wikipedia, or any one of my reference books before answering a simple definition from the hip.

Rap

I was referring to Engineer's post not the initisl post.