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and these , , 
and , 

Cross multiple the first two:

and x=r*cos(q), y=r*cos(q), and y/x=tan(q), r² = x² + y²

y = x²

Replace y by r*sin(q)
Replace x by r*cos(q)

r*sin(q) = [r*cos(q)]²

r*sin(q) = r²cos²(q)

Divide both sides by r

sin(q) = r*cos²(q)

Swap sides:

r*cos²(q) = sin(q)
     
     sin(q)
r = —————————
     cos²(q)

You can leave it like that. Or else, write as

       sin(q)
r = ——————————————
     cos(q)*cos(q)


    sin(q)     1
r = —————— × ——————  
    cos(q)   cos(q)


r = tan(q)*sec(q)

Edwin