What is (1+cos(x)+isin(x))^2 ?

The solution to (1+cos(x)+isin(x))^2 is (2cos(x)+cos^2(x)-sin^2(x)+1)+(2sin(x)+sin(2x))i

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simplify (1+cos(x)+isin(x))^2

simplify

(1 + cos (x) + i sin (x))^{2}

(2 cos (x) + cos^{2} (x) - sin^{2} (x) + 1) + (2 sin (x) + sin (2 x)) i

Solution steps

(1 + cos (x) + i sin (x))^{2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

(1 + cos (x) + i sin (x))^{2} = (1 + cos (x) + i sin (x)) (1 + cos (x) + i sin (x))

= (1 + cos (x) + i sin (x)) (1 + cos (x) + i sin (x))

Distribute parentheses

= 1 \cdot 1 + 1 \cdot cos (x) + 1 i sin (x) + cos (x) \cdot 1 + cos (x) cos (x) + cos (x) i sin (x) + i sin (x) \cdot 1 + i sin (x) cos (x) + i sin (x) i sin (x)

Simplify

1 \cdot 1 + 1 \cdot cos (x) + 1 i sin (x) + cos (x) \cdot 1 + cos (x) cos (x) + cos (x) i sin (x) + i sin (x) \cdot 1 + i sin (x) cos (x) + i sin (x) i sin (x) : 2 cos (x) + 2 sin (x) i + cos^{2} (x) + 2 cos (x) sin (x) i - sin^{2} (x) + 1

= 2 cos (x) + 2 sin (x) i + cos^{2} (x) + 2 cos (x) sin (x) i - sin^{2} (x) + 1

Rewrite in standard complex form:

(2 cos (x) + cos^{2} (x) - sin^{2} (x) + 1) + (2 sin (x) + 2 cos (x) sin (x)) i

= (2 cos (x) + cos^{2} (x) - sin^{2} (x) + 1) + (2 sin (x) + 2 cos (x) sin (x)) i

Simplify

2 sin (x) + 2 cos (x) sin (x) : 2 sin (x) + sin (2 x)

= (2 cos (x) + cos^{2} (x) - sin^{2} (x) + 1) + (2 sin (x) + sin (2 x)) i

- 7 (5 x + 42 y = - 69)

tan^{2} (x) (cot^{2} (x) - cos^{2} (x))

\frac{1}{x \cdot ( x - 1 )}

(4 - \frac{x}{3})^{5}

10 (1 - i w) (5 + i w) (10 - i w) (50 - i w)

What is (1+cos(x)+isin(x))^2 ?
The solution to (1+cos(x)+isin(x))^2 is (2cos(x)+cos^2(x)-sin^2(x)+1)+(2sin(x)+sin(2x))i