Question

If(cos theta+ i sin theta) sq = x+iy p:t x sq +y sq=1​

Answer 1

EXPLAINATION

GIVEN

$⇒ x + iy = (cosθ + isinθ)²$

FORMULA

${e}^{i \theta} = \cos( \theta) + i \sin( \theta )$

PROOF

$x + iy = (cosθ + isinθ)²$

$x + iy = (eⁱᶿ)²$

$x + iy = eⁱ⁽²ᶿ⁾------(1)$

Taking conjugate on both side

$x - iy = e⁻ⁱ⁽²ᶿ⁾------(2)$

Multiplying (1) and (2), we get

$(x + iy)(x - iy) = eⁱ⁽²ᶿ⁾.e⁻ⁱ⁽²ᶿ⁾$

$(x² + y²) = eⁱ⁽²ᶿ⁾⁻ⁱ⁽²ᶿ⁾$

$(x² + y²) = e⁰$

$(x² + y²) = 1$

Hence Proved.

OR

$x + iy = (cosθ + isinθ)²$

Taking conjugate on both side

$x - iy = (cosθ - isinθ)²$

Multiplying, we get

$(x + iy)(x - iy)$

$= (cosθ + isinθ)²(cosθ- isinθ)²$

$= [(cosθ + isinθ)(cosθ- isinθ)]²$

$= [(cos²θ + sin²θ)]²$

$= [1]² = 1$

Hence Proved