Trigonometry Examples

cos (6 x)

$\cos (6 x)$

Step 1

A good method to expand

cos (6 x)

$\cos (6 x)$ is by using De Moivre's theorem

(r {(cos (x) + i \cdot sin (x))}^{n} = r^{n} (cos (n x) + i \cdot sin (n x)))

$(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . When

r = 1

$r = 1$ ,

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$

Step 2

Expand the right hand side of

cos (n x) + i \cdot sin (n x) = {(cos (x) + i \cdot sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ using the binomial theorem.

Expand:

{(cos (x) + i \cdot sin (x))}^{6}

${(\cos (x) + i \cdot \sin (x))}^{6}$

Step 3

Use the Binomial Theorem.

{cos}^{6} (x) + 6 {cos}^{5} (x) (i sin (x)) + 15 {cos}^{4} (x) {(i sin (x))}^{2} + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) (i \sin (x)) + 15 \cos^{4} (x) {(i \sin (x))}^{2} + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 {cos}^{4} (x) (i^{2} {sin}^{2} (x)) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cos^{4} (x) (i^{2} \sin^{2} (x)) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.2

Rewrite using the commutative property of multiplication.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 \cdot i^{2} {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot i^{2} \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.3

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) + 15 \cdot - 1 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) + 15 \cdot - 1 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.4

Multiply

15

$15$ by

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) {(i sin (x))}^{3} + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) {(i \sin (x))}^{3} + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.5

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 {cos}^{3} (x) (i^{3} {sin}^{3} (x)) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cos^{3} (x) (i^{3} \sin^{3} (x)) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.6

Rewrite using the commutative property of multiplication.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot i^{3} {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot i^{3} \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.7

Factor out

i^{2}

$i^{2}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (i^{2} \cdot i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (i^{2} \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.8

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (- 1 \cdot i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- 1 \cdot i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.9

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) + 20 \cdot (- i) {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) + 20 \cdot (- i) \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.10

Multiply

- 1

$- 1$ by

20

$20$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {(i sin (x))}^{4} + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) {(i \sin (x))}^{4} + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.11

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) (i^{4} {sin}^{4} (x)) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) (i^{4} \sin^{4} (x)) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.12

Rewrite using the commutative property of multiplication.

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot i^{4} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot i^{4} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.13

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

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Step 4.1.13.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot {(i^{2})}^{2} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(i^{2})}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.13.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot {(- 1)}^{2} {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot {(- 1)}^{2} \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.13.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot 1 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot 1 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 \cdot 1 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cdot 1 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.14

Multiply

15

$15$ by

1

$1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) {(i sin (x))}^{5} + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) {(i \sin (x))}^{5} + {(i \sin (x))}^{6}$

Step 4.1.15

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i^{5} {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{5} \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.16

Factor out

i^{4}

$i^{4}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i^{4} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i^{4} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.17

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

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Step 4.1.17.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) ({(i^{2})}^{2} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(i^{2})}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.17.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) ({(- 1)}^{2} i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) ({(- 1)}^{2} i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.17.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (1 i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (1 i \sin^{5} (x)) + {(i \sin (x))}^{6}$

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (1 i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (1 i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.18

Multiply

i

$i$ by

1

$1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) (i {sin}^{5} (x)) + {(i sin (x))}^{6}

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) (i \sin^{5} (x)) + {(i \sin (x))}^{6}$

Step 4.1.19

Apply the product rule to

i sin (x)

$i \sin (x)$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + i^{6} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{6} \sin^{6} (x)$

Step 4.1.20

Factor out

i^{4}

$i^{4}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + i^{4} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{4} i^{2} \sin^{6} (x)$

Step 4.1.21

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

Tap for more steps...

Step 4.1.21.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + {(i^{2})}^{2} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(i^{2})}^{2} i^{2} \sin^{6} (x)$

Step 4.1.21.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

{cos}^{6} (x) + 6 {cos}^{5} (x) i sin (x) - 15 {cos}^{4} (x) {sin}^{2} (x) - 20 i {cos}^{3} (x) {sin}^{3} (x) + 15 {cos}^{2} (x) {sin}^{4} (x) + 6 cos (x) i {sin}^{5} (x) + {(- 1)}^{2} i^{2} {sin}^{6} (x)

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + {(- 1)}^{2} i^{2} \sin^{6} (x)$

Step 4.1.21.3

Raise

$- 1$ to the power of

$2$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + 1 i^{2} \sin^{6} (x)$

Step 4.1.22

Multiply

$i^{2}$ by

$1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) + i^{2} \sin^{6} (x)$

Step 4.1.23

Rewrite

$i^{2}$ as

$- 1$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - 1 \sin^{6} (x)$

Step 4.1.24

Rewrite

$- 1 \sin^{6} (x)$ as

$- \sin^{6} (x)$ .

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$

Step 4.2

Reorder factors in

$\cos^{6} (x) + 6 \cos^{5} (x) i \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 \cos (x) i \sin^{5} (x) - \sin^{6} (x)$ .

$\cos^{6} (x) + 6 i \cos^{5} (x) \sin (x) - 15 \cos^{4} (x) \sin^{2} (x) - 20 i \cos^{3} (x) \sin^{3} (x) + 15 \cos^{2} (x) \sin^{4} (x) + 6 i \cos (x) \sin^{5} (x) - \sin^{6} (x)$

Step 5

Take out the expressions with the imaginary part, which are equal to

$\cos (6 x)$ . Remove the imaginary number

$i$ .

$\cos (6 x) = \cos^{6} (x) - 15 \cos^{4} (x) \sin^{2} (x) + 15 \cos^{2} (x) \sin^{4} (x) - \sin^{6} (x)$

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