What is the general solution for 4cos^2(x)+4sin^2(x)-5=0 ?

The general solution for 4cos^2(x)+4sin^2(x)-5=0 is No Solution for x\in\mathbb{R}

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4cos^2(x)+4sin^2(x)-5=0

Solution

4 cos^{2} (x) + 4 sin^{2} (x) - 5 = 0

Solution

No Solution for x \in R

Solution steps

4 cos^{2} (x) + 4 sin^{2} (x) - 5 = 0

Subtract

4 sin^{2} (x)

from both sides

4 cos^{2} (x) - 5 = - 4 sin^{2} (x)

Square both sides

(4 cos^{2} (x) - 5)^{2} = (- 4 sin^{2} (x))^{2}

Subtract

(- 4 sin^{2} (x))^{2}

from both sides

(4 cos^{2} (x) - 5)^{2} - 16 sin^{4} (x) = 0

Rewrite using trig identities

(- 5 + 4 cos^{2} (x))^{2} - 16 sin^{4} (x)

Use the Pythagorean identity:

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

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

= (- 5 + 4 (1 - sin^{2} (x)))^{2} - 16 sin^{4} (x)

Simplify

(- 5 + 4 (1 - sin^{2} (x)))^{2} - 16 sin^{4} (x) : 8 sin^{2} (x) + 1

(- 5 + 4 (1 - sin^{2} (x)))^{2} - 16 sin^{4} (x)

Expand

- 5 + 4 (1 - sin^{2} (x)) : - 4 sin^{2} (x) - 1

- 5 + 4 (1 - sin^{2} (x))

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= - 5 + 4 - 4 sin^{2} (x)

Add/Subtract the numbers:

- 5 + 4 = - 1

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

= (- 4 sin^{2} (x) - 1)^{2} - 16 sin^{4} (x)

(- 4 sin^{2} (x) - 1)^{2} : 16 sin^{4} (x) + 8 sin^{2} (x) + 1

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = - 4 sin^{2} (x), b = 1

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

Simplify

(- 4 sin^{2} (x))^{2} - 2 (- 4 sin^{2} (x)) \cdot 1 + 1^{2} : 16 sin^{4} (x) + 8 sin^{2} (x) + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Apply rule

- (- a) = a

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

(- 4 sin^{2} (x))^{2} = 16 sin^{4} (x)

(- 4 sin^{2} (x))^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 4 sin^{2} (x))^{2} = (4 sin^{2} (x))^{2}

= (4 sin^{2} (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (sin^{2} (x))^{2}

(sin^{2} (x))^{2} : sin^{4} (x)

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= sin^{2 \cdot 2} (x)

Multiply the numbers:

2 \cdot 2 = 4

= sin^{4} (x)

= 4^{2} sin^{4} (x)

4^{2} = 16

= 16 sin^{4} (x)

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

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

Multiply the numbers:

2 \cdot 4 \cdot 1 = 8

= 8 sin^{2} (x)

= 16 sin^{4} (x) + 8 sin^{2} (x) + 1

= 16 sin^{4} (x) + 8 sin^{2} (x) + 1

= 16 sin^{4} (x) + 8 sin^{2} (x) + 1 - 16 sin^{4} (x)

Simplify

16 sin^{4} (x) + 8 sin^{2} (x) + 1 - 16 sin^{4} (x) : 8 sin^{2} (x) + 1

16 sin^{4} (x) + 8 sin^{2} (x) + 1 - 16 sin^{4} (x)

Group like terms

= 16 sin^{4} (x) + 8 sin^{2} (x) - 16 sin^{4} (x) + 1

Add similar elements:

16 sin^{4} (x) - 16 sin^{4} (x) = 0

= 8 sin^{2} (x) + 1

= 8 sin^{2} (x) + 1

= 8 sin^{2} (x) + 1

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

Solve by substitution

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

Let:

sin (x) = u

1 + 8 u^{2} = 0

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

1 + 8 u^{2} = 0

Move

1

to the right side

1 + 8 u^{2} = 0

Subtract

1

from both sides

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

Simplify

8 u^{2} = - 1

8 u^{2} = - 1

Divide both sides by

8

8 u^{2} = - 1

Divide both sides by

8

\frac{8 u ^{2}}{8} = \frac{- 1}{8}

Simplify

u^{2} = - \frac{1}{8}

u^{2} = - \frac{1}{8}

For

x^{2} = f (a)

the solutions are

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

u = - \frac{1}{8}, u = - - \frac{1}{8}

Simplify

- \frac{1}{8} : i \frac{2}{4}

- \frac{1}{8}

Apply radical rule:

- a = - 1 a

- \frac{1}{8} = - 1 \frac{1}{8}

= - 1 \frac{1}{8}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{8} = \frac{1}{8}

= i \frac{1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{1}{2 2}

Apply rule

1 = 1

= i \frac{1}{2 2}

\frac{1}{2 2} = \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= i \frac{2}{4}

Rewrite

i \frac{2}{4}

in standard complex form:

\frac{2}{4} i

i \frac{2}{4}

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2}{2 ^{2}}

Cancel

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{1}{2 2}

= i \frac{1}{2 2}

Multiply fractions:

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

= \frac{1 i}{2 2}

Multiply:

1 i = i

= \frac{i}{2 2}

\frac{1}{2 2} = \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4} i

= \frac{2}{4} i

Simplify

- - \frac{1}{8} : - i \frac{2}{4}

- - \frac{1}{8}

Simplify

- \frac{1}{8} : i \frac{1}{2 2}

- \frac{1}{8}

Apply radical rule:

- a = - 1 a

- \frac{1}{8} = - 1 \frac{1}{8}

= - 1 \frac{1}{8}

Apply imaginary number rule:

- 1 = i

= i \frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{1}{8} = \frac{1}{8}

= i \frac{1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{1}{2 2}

= - i \frac{1}{2 2}

Apply rule

1 = 1

= - i \frac{1}{2 2}

\frac{1}{2 2} = \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= - \frac{2}{4} i

u = i \frac{2}{4}, u = - i \frac{2}{4}

Substitute back

u = sin (x)

sin (x) = i \frac{2}{4}, sin (x) = - i \frac{2}{4}

sin (x) = i \frac{2}{4}, sin (x) = - i \frac{2}{4}

sin (x) = i \frac{2}{4} :

No Solution

sin (x) = i \frac{2}{4}

No Solution

sin (x) = - i \frac{2}{4} :

No Solution

sin (x) = - i \frac{2}{4}

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

4 cos^{2} (x) + 4 sin^{2} (x) - 5 = 0

Remove the ones that don't agree with the equation.

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos^2(x)+4sin^2(x)-5=0 ?
The general solution for 4cos^2(x)+4sin^2(x)-5=0 is No Solution for x\in\mathbb{R}