What is (1+sin(x)+icos(x))^2 ?

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

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simplify (1+sin(x)+icos(x))^2

simplify

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Distribute parentheses

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

Simplify

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

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

Rewrite in standard complex form:

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

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

Simplify

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

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

(1 + csc (x))^{2}

\frac{2 x ^{3} + 3 x ^{2} - 15 x - 8}{( x + 2 ) ( x ^{3} - 3 x + 2 )}

\frac{d}{d v} ((p) (p + \frac{9.5}{v ^{2}}) (v - 4.4))

(tan (x) + c t an x) \cdot sin (2 x)

(x + 3) (x + 2)^{\frac{1}{2}}

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