Thread: Identity and Curve

Arghhh I'm really stuck on these 2 problems. I never seem to be able to get them out. Any help would be greatsly appreciated.


Prove the identity:

cosx/1-tanx + sinx/1-cotx = sinx + cosx


and also:

Eliminate x from the equations below:

y = 3 - 2sinx
x = 4 + 5cotx

Hello, danco!

Prove the identity: .cos x/(1 - tan x) + sin x/(1 - cot x) .= .sin x + cos x

. . . . . . . . . . cos x . . . . .sin x
We have: . ----------- + ------------
. . . . . . . . . . .sin x . . . . . cos x
. . . . . . . . 1 - ------ . . 1 - -------
. . . . . . . . . . .cos x . . . . .sin x


Multiply top and bottom of the first fraction by cos x.
Multiply top and bottom of the second fraction by sin x.

. . . . . . . . . . . . . cos²x . . . . . . .sin²x . . . . . . . . .cos²x . . . . . . sin²x
And we have: . --------------- + -------------- . = . --------------- - ---------------
. . . . . . . . . . . cos x - sin x . .sin x - cos x . . . . cos x - sin x . .cosx - sin x


. . . . . . cos²x - sin²x . . . . (cos x - sin x)(cos x + sin x)
. . . = . ---------------- . = . ---------------------------------- . = . cos x + sin x
. . . . . . cos x - sin x . . . . . . . . . .cos x - sin x

I suspect there are typos in this one.

Eliminate θ from the equations below:

. . y .= .3 - 2·sinθ
. . x .= .4 + 5·cotθ . ← I bet this is cosine

From the first equation: .y .= .3 - 2·sinθ . → . sinθ .= .(3 - y)/2 .[1]

From the second equation: .x .= .4 + 5·cosθ . → . cosθ .= .(x - 4)/5 .[2]


Square [1]: .sin²θ .= .(3 - y)²/4

Square [2]: .cos²θ .= .(x - 4)²/25


Add: .(3 - y)²/4 + (x - 4)²/25 .= .sin²θ + cos²θ .= .1


. . . . . . . . . . . . . . . . . . (x - 4)² . (y - 3)²
And we have an ellipse: . --------- + -------- .= .1
. . . . . . . . . . . . . . . . . . . .25 . . . . . 4