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Popular >

sin^2(x)cos^2(x)=((1-cos^4(x)))/8

Solution

sin^{2} (x) cos^{2} (x) = \frac{( 1 - cos ^{4} ( x ) )}{8}

Solution

x = 1.18319 \dots + 2 πn, x = 2 π - 1.18319 \dots + 2 πn, x = 1.95839 \dots + 2 πn, x = - 1.95839 \dots + 2 πn, x = 2 πn, x = π + 2 πn

Degrees

x = 67.79234 \dots^{\circ} + 36 0^{\circ} n, x = 292.20765 \dots^{\circ} + 36 0^{\circ} n, x = 112.20765 \dots^{\circ} + 36 0^{\circ} n, x = - 112.20765 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) cos^{2} (x) = \frac{( 1 - cos ^{4} ( x ) )}{8}

Subtract

\frac{1 - cos ^{4} ( x )}{8}

from both sides

sin^{2} (x) cos^{2} (x) - \frac{1 - cos ^{4} ( x )}{8} = 0

Simplify

sin^{2} (x) cos^{2} (x) - \frac{1 - cos ^{4} ( x )}{8} : \frac{8 sin ^{2} ( x ) cos ^{2} ( x ) - 1 + cos ^{4} ( x )}{8}

sin^{2} (x) cos^{2} (x) - \frac{1 - cos ^{4} ( x )}{8}

Convert element to fraction:

sin^{2} (x) cos^{2} (x) = \frac{sin ^{2} ( x ) cos ^{2} ( x ) 8}{8}

= \frac{sin ^{2} ( x ) cos ^{2} ( x ) \cdot 8}{8} - \frac{1 - cos ^{4} ( x )}{8}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ^{2} ( x ) cos ^{2} ( x ) \cdot 8 - ( 1 - cos ^{4} ( x ) )}{8}

Expand

sin^{2} (x) cos^{2} (x) \cdot 8 - (1 - cos^{4} (x)) : sin^{2} (x) cos^{2} (x) \cdot 8 - 1 + cos^{4} (x)

sin^{2} (x) cos^{2} (x) \cdot 8 - (1 - cos^{4} (x))

= 8 sin^{2} (x) cos^{2} (x) - (1 - cos^{4} (x))

- (1 - cos^{4} (x)) : - 1 + cos^{4} (x)

- (1 - cos^{4} (x))

Distribute parentheses

= - (1) - (- cos^{4} (x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + cos^{4} (x)

= sin^{2} (x) cos^{2} (x) \cdot 8 - 1 + cos^{4} (x)

= \frac{8 sin ^{2} ( x ) cos ^{2} ( x ) - 1 + cos ^{4} ( x )}{8}

\frac{8 sin ^{2} ( x ) cos ^{2} ( x ) - 1 + cos ^{4} ( x )}{8} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

8 sin^{2} (x) cos^{2} (x) - 1 + cos^{4} (x) = 0

Rewrite using trig identities

- 1 + cos^{4} (x) + 8 cos^{2} (x) sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 1 + cos^{4} (x) + 8 cos^{2} (x) (1 - cos^{2} (x))

Simplify

- 1 + cos^{4} (x) + 8 cos^{2} (x) (1 - cos^{2} (x)) : - 7 cos^{4} (x) + 8 cos^{2} (x) - 1

- 1 + cos^{4} (x) + 8 cos^{2} (x) (1 - cos^{2} (x))

Expand

8 cos^{2} (x) (1 - cos^{2} (x)) : 8 cos^{2} (x) - 8 cos^{4} (x)

8 cos^{2} (x) (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 8 cos^{2} (x), b = 1, c = cos^{2} (x)

= 8 cos^{2} (x) \cdot 1 - 8 cos^{2} (x) cos^{2} (x)

= 8 \cdot 1 \cdot cos^{2} (x) - 8 cos^{2} (x) cos^{2} (x)

Simplify

8 \cdot 1 \cdot cos^{2} (x) - 8 cos^{2} (x) cos^{2} (x) : 8 cos^{2} (x) - 8 cos^{4} (x)

8 \cdot 1 \cdot cos^{2} (x) - 8 cos^{2} (x) cos^{2} (x)

8 \cdot 1 \cdot cos^{2} (x) = 8 cos^{2} (x)

8 \cdot 1 \cdot cos^{2} (x)

Multiply the numbers:

8 \cdot 1 = 8

= 8 cos^{2} (x)

8 cos^{2} (x) cos^{2} (x) = 8 cos^{4} (x)

8 cos^{2} (x) cos^{2} (x)

Apply exponent rule:

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

cos^{2} (x) cos^{2} (x) = cos^{2 + 2} (x)

= 8 cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= 8 cos^{4} (x)

= 8 cos^{2} (x) - 8 cos^{4} (x)

= 8 cos^{2} (x) - 8 cos^{4} (x)

= - 1 + cos^{4} (x) + 8 cos^{2} (x) - 8 cos^{4} (x)

Simplify

- 1 + cos^{4} (x) + 8 cos^{2} (x) - 8 cos^{4} (x) : - 7 cos^{4} (x) + 8 cos^{2} (x) - 1

- 1 + cos^{4} (x) + 8 cos^{2} (x) - 8 cos^{4} (x)

Group like terms

= cos^{4} (x) + 8 cos^{2} (x) - 8 cos^{4} (x) - 1

Add similar elements:

cos^{4} (x) - 8 cos^{4} (x) = - 7 cos^{4} (x)

= - 7 cos^{4} (x) + 8 cos^{2} (x) - 1

= - 7 cos^{4} (x) + 8 cos^{2} (x) - 1

= - 7 cos^{4} (x) + 8 cos^{2} (x) - 1

- 1 - 7 cos^{4} (x) + 8 cos^{2} (x) = 0

Solve by substitution

- 1 - 7 cos^{4} (x) + 8 cos^{2} (x) = 0

Let:

cos (x) = u

- 1 - 7 u^{4} + 8 u^{2} = 0

- 1 - 7 u^{4} + 8 u^{2} = 0 : u = \frac{1}{7}, u = - \frac{1}{7}, u = 1, u = - 1

- 1 - 7 u^{4} + 8 u^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 7 u^{4} + 8 u^{2} - 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 7 v^{2} + 8 v - 1 = 0

Solve

- 7 v^{2} + 8 v - 1 = 0 : v = \frac{1}{7}, v = 1

- 7 v^{2} + 8 v - 1 = 0

Solve with the quadratic formula

- 7 v^{2} + 8 v - 1 = 0

Quadratic Equation Formula:

For

a = - 7, b = 8, c = - 1

v_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 7 ) ( - 1 )}{2 ( - 7 )}

v_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 7 ) ( - 1 )}{2 ( - 7 )}

8^{2} - 4 (- 7) (- 1) = 6

8^{2} - 4 (- 7) (- 1)

Apply rule

- (- a) = a

= 8^{2} - 4 \cdot 7 \cdot 1

Multiply the numbers:

4 \cdot 7 \cdot 1 = 28

= 8^{2} - 28

8^{2} = 64

= 64 - 28

Subtract the numbers:

64 - 28 = 36

= 36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

v_{1, 2} = \frac{- 8 \pm 6}{2 ( - 7 )}

Separate the solutions

v_{1} = \frac{- 8 + 6}{2 ( - 7 )}, v_{2} = \frac{- 8 - 6}{2 ( - 7 )}

v = \frac{- 8 + 6}{2 ( - 7 )} : \frac{1}{7}

\frac{- 8 + 6}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 6}{- 2 \cdot 7}

Add/Subtract the numbers:

- 8 + 6 = - 2

= \frac{- 2}{- 2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 2}{- 14}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2}{14}

Cancel the common factor:

2

= \frac{1}{7}

v = \frac{- 8 - 6}{2 ( - 7 )} : 1

\frac{- 8 - 6}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 6}{- 2 \cdot 7}

Subtract the numbers:

- 8 - 6 = - 14

= \frac{- 14}{- 2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 14}{- 14}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{14}{14}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

v = \frac{1}{7}, v = 1

v = \frac{1}{7}, v = 1

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{1}{7} : u = \frac{1}{7}, u = - \frac{1}{7}

u^{2} = \frac{1}{7}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{7}, u = - \frac{1}{7}

Solve

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

The solutions are

u = \frac{1}{7}, u = - \frac{1}{7}, u = 1, u = - 1

Substitute back

u = cos (x)

cos (x) = \frac{1}{7}, cos (x) = - \frac{1}{7}, cos (x) = 1, cos (x) = - 1

cos (x) = \frac{1}{7}, cos (x) = - \frac{1}{7}, cos (x) = 1, cos (x) = - 1

cos (x) = \frac{1}{7} : x = arccos (\frac{1}{7}) + 2 πn, x = 2 π - arccos (\frac{1}{7}) + 2 πn

cos (x) = \frac{1}{7}

Apply trig inverse properties

cos (x) = \frac{1}{7}

General solutions for

cos (x) = \frac{1}{7}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{7}) + 2 πn, x = 2 π - arccos (\frac{1}{7}) + 2 πn

x = arccos (\frac{1}{7}) + 2 πn, x = 2 π - arccos (\frac{1}{7}) + 2 πn

cos (x) = - \frac{1}{7} : x = arccos (- \frac{1}{7}) + 2 πn, x = - arccos (- \frac{1}{7}) + 2 πn

cos (x) = - \frac{1}{7}

Apply trig inverse properties

cos (x) = - \frac{1}{7}

General solutions for

cos (x) = - \frac{1}{7}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{7}) + 2 πn, x = - arccos (- \frac{1}{7}) + 2 πn

x = arccos (- \frac{1}{7}) + 2 πn, x = - arccos (- \frac{1}{7}) + 2 πn

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

General solutions for

cos (x) = - 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = arccos (\frac{1}{7}) + 2 πn, x = 2 π - arccos (\frac{1}{7}) + 2 πn, x = arccos (- \frac{1}{7}) + 2 πn, x = - arccos (- \frac{1}{7}) + 2 πn, x = 2 πn, x = π + 2 πn

Show solutions in decimal form

x = 1.18319 \dots + 2 πn, x = 2 π - 1.18319 \dots + 2 πn, x = 1.95839 \dots + 2 πn, x = - 1.95839 \dots + 2 πn, x = 2 πn, x = π + 2 πn

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