Given:

The given curves are x=cosθ+sin2θ and y=sinθ+cos2θ . The parametric value is θ=0 .

Calculation:

Find the x and y value at θ=0 .

  x=cosθ+sin2θ=cos(0)+sin(2⋅0)=1

  y=sinθ+cos2θ=sin(0)+cos(2⋅0)=1

Find the slope.

Apply formula dydx=dydtdxdt .

  dydθ=ddθ(sinθ+cos2θ)

Apply sum rule (f+g)'=f'+g' .

  ⇒dydθ=ddθ(sinθ)+ddθ(cos2θ)

Use derivative rule ddx(sinθ)=cosθ and ddx(cosθ)=−sinθ .

  ⇒dydθ=cosθ−sin2θ⋅2⇒dydθ=cosθ−2sin2θ

  dxdθ=ddθ(cosθ+sin2θ)

Apply sum rule (f+g)'=f'+g' .

  ⇒dxdθ=ddθ(cosθ)+ddθ(sin2θ)

Use derivative rule ddx(sinθ)=cosθ and ddx(cosθ)=−sinθ .

  ⇒dxdθ=−sinθ+2cos2θ

  dydx(slope)=cosθ−2sin2θ−sinθ+2cos2θ

At the value θ=0 .

  dydx(slope)=cosθ−2sin2θ−sinθ+2cos2θ=cos(0)−2sin(2⋅0)−sin(0)+2cos(2⋅0)=12

Use point-slope form for the equation of tangent line.

  y−y1=m(x−x1)y1=1,x1=1,m=12y−1=12(x−1)⇒y−1=x2−12⇒y−1+1=x2−12+1⇒y=x2+12

Hence theequation of the tangent line is y=x2+12 .