What is the general solution for cos^2(x)=1-cos^2(x),[360] ?

The general solution for cos^2(x)=1-cos^2(x),[360] is No Solution for x\in\mathbb{R}

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Popular >

cos^2(x)=1-cos^2(x),[360]

Solution

cos^{2} (x) = 1 - cos^{2} (x), [36 0^{\circ}]

Solution

No Solution for x \in R

Solution steps

cos^{2} (x) = 1 - cos^{2} (x), [36 0^{\circ}]

Solve by substitution

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

Let:

cos (x) = u

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

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

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

Move

u^{2}

to the left side

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

Add

u^{2}

to both sides

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

Simplify

2 u^{2} = 1

2 u^{2} = 1

Divide both sides by

2

2 u^{2} = 1

Divide both sides by

2

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{1}{2}, [36 0^{\circ}] :

No Solution

cos (x) = \frac{1}{2}, [36 0^{\circ}]

Apply trig inverse properties

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x = arccos (\frac{1}{2}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{1}{2}) + 36 0^{\circ} n

x = arccos (\frac{1}{2}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{1}{2}) + 36 0^{\circ} n

Solutions for the range

[36 0^{\circ}]

No Solution

cos (x) = - \frac{1}{2}, [36 0^{\circ}] :

No Solution

cos (x) = - \frac{1}{2}, [36 0^{\circ}]

Apply trig inverse properties

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

x = arccos (- \frac{1}{2}) + 36 0^{\circ} n, x = - arccos (- \frac{1}{2}) + 36 0^{\circ} n

x = arccos (- \frac{1}{2}) + 36 0^{\circ} n, x = - arccos (- \frac{1}{2}) + 36 0^{\circ} n

Solutions for the range

[36 0^{\circ}]

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)=1-cos^2(x),[360] ?
The general solution for cos^2(x)=1-cos^2(x),[360] is No Solution for x\in\mathbb{R}