Question

cos theta +isin theta)^2= x+iy ,prove x^2+y^2 =1

Answer 1

hope this will help you. thank you for asking a good question.

Answer 2

Concept

There are certain trigonometric identities we must know in order to solve this question. These are-

$cosx^{2} + sinx^{2} = 1$

$sin2x = 2sinxcosx$

$cos2x = cosx^{2} - sinx^{2}$

Given

(cos ∅ + i sin ∅)^2 = x + i y

Find

we are asked to prove that $x^{2} + y^{2} = 1$

Solution

We have,

(cos ∅ + i sin ∅)^2 = x + i y

using $(a+b)^{2}$  formula, we get

(cos∅)^2 + (i sin∅)^2 + 2sin∅cos∅ = x + iy

also, we know that $i = \sqrt{-1}$

⇒ $i^{2} = -1$

Therefore, by replacing the value of $i^{2}$ in the equation we get-

(cos∅)^2 - (sin∅)^2 + 2isin∅cos∅ = x + iy

Now as per identity, (cos∅)^2 - (sin∅)^2 = cos2∅

⇒ cos2∅ + 2isin∅cos∅ = x + iy

Also, sin2∅ = 2sin∅cos∅

⇒ cos2∅ + isin2∅ = x + iy

Now, comparing the real and imaginary part we get,

cos2∅ = x

& isin2∅ = iy

or sin2∅ = y

Thus, $x^{2} + y^{2} =$ (cos2∅)^2 + (sin2∅)^2

⇒ $x^{2} + y^{2} = 1$

Hence proved.  

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