We have,
x=cos4θ ….. (1)
y=sin4θ …… (2)
On differentiation (1) and (2) to, we get,
dxdθ=−4cos3θsinθ
dydθ=4sin3θcosθ
Then,
dydx=dydθdxdθ=4sin3θcosθ−4cos3θsinθ
dydx=−tan2θ
Now equation of tangent is
y−y1=dydx(x−x1)
y−sin4θ=−tan2θ(x−cos4θ)
y−sin4θ=−sin2θcos2θ(x−cos4θ)
ycos2θ−sin4θcos2θ=−xsin2θ+sin2θcos4θ
xsin2θ+ycos2θ=sin2θcos4θ+sin4θcos2θ
xsin2θ+ycos2θ=sin2θcos2θ(cos2θ+sin2θ)∴cos2θ+sin2θ=1
xsin2θ+ycos2θ=sin2θcos2θ
On divide both side sin2θcos2θ
So,
xsin2θsin2θcos2θ+ycos2θsin2θcos2θ=sin2θcos2θsin2θcos2θ
xcos2θ+ysin2θ=1
xsec2θ+ycsc2θ=1
It is equation of tangent