1 Find the polar equation that has the same graph as the giv

1. Find the polar equation that has the same graph as the given rectangular equation: x^3 + y^3 -xy = 0

2. Prove this identity: 1 + cot^3 t/ 1 + cot t = csc^2 t - cot t

3. Prove this identity: sin x + 1 / cosx + cotx = tan x

Solution

1)x^3 + y^3 -xy = 0

in polar coordinates

x=rcos, y=rsin,x²+y²=r²

(rcos)³+(rsin)³-(rcos rsin)=0

r²(r(cos)³+r(sin)³-cossin)=0

(r(cos)³+r(sin)³-cossin)=0

2)(1 + cot³t)/ (1 + cott)

(1³ + cot³t)/ (1 + cott)

a³+b³=(a+b)(a²-ab+b²)

=(1 + cott)(1-cott+cot²t)/ (1 + cott)

cancel out (1 + cott)

=(1-cott+cot²t)

=(1+cot²t-cott)

=csc²t -cott

3)(sin x + 1)/ (cosx + cotx )

(sin x + 1)/ (cosx + (cosx/sinx) )

(sin x + 1)/ (sinxcosx + cosx )/sinx

(sin x + 1)sinx/ (sinxcosx + cosx )

(sin x + 1)sinx/ (cosx (sinx+1))

cancel out sinx +1

=sinx/cosx

=tanx