What is the general solution for cos^2(x)=((8k-7))/7 ?

The general solution for cos^2(x)=((8k-7))/7 is

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cos^2(x)=((8k-7))/7

cos^{2} (x) = \frac{( 8 k - 7 )}{7}

x = arccos (\frac{8 k - 7}{7}) + 2 πn, x = - arccos (\frac{8 k - 7}{7}) + 2 πn, x = arccos (- \frac{8 k - 7}{7}) + 2 πn, x = - arccos (- \frac{8 k - 7}{7}) + 2 πn

Solution steps

cos^{2} (x) = \frac{( 8 k - 7 )}{7}

Solve by substitution

cos^{2} (x) = \frac{8 k - 7}{7}

Let:

cos (x) = u

u^{2} = \frac{8 k - 7}{7}

u^{2} = \frac{8 k - 7}{7} : u = \frac{8 k - 7}{7}, u = - \frac{8 k - 7}{7}

u^{2} = \frac{8 k - 7}{7}

For

x^{2} = f (a)

the solutions are

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

u = \frac{8 k - 7}{7}, u = - \frac{8 k - 7}{7}

Substitute back

u = cos (x)

cos (x) = \frac{8 k - 7}{7}, cos (x) = - \frac{8 k - 7}{7}

cos (x) = \frac{8 k - 7}{7}, cos (x) = - \frac{8 k - 7}{7}

cos (x) = \frac{8 k - 7}{7} : x = arccos (\frac{8 k - 7}{7}) + 2 πn, x = - arccos (\frac{8 k - 7}{7}) + 2 πn

cos (x) = \frac{8 k - 7}{7}

Apply trig inverse properties

cos (x) = \frac{8 k - 7}{7}

General solutions for

cos (x) = \frac{8 k - 7}{7}

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

x = arccos (\frac{8 k - 7}{7}) + 2 πn, x = - arccos (\frac{8 k - 7}{7}) + 2 πn

x = arccos (\frac{8 k - 7}{7}) + 2 πn, x = - arccos (\frac{8 k - 7}{7}) + 2 πn

cos (x) = - \frac{8 k - 7}{7} : x = arccos (- \frac{8 k - 7}{7}) + 2 πn, x = - arccos (- \frac{8 k - 7}{7}) + 2 πn

cos (x) = - \frac{8 k - 7}{7}

Apply trig inverse properties

cos (x) = - \frac{8 k - 7}{7}

General solutions for

cos (x) = - \frac{8 k - 7}{7}

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

x = arccos (- \frac{8 k - 7}{7}) + 2 πn, x = - arccos (- \frac{8 k - 7}{7}) + 2 πn

x = arccos (- \frac{8 k - 7}{7}) + 2 πn, x = - arccos (- \frac{8 k - 7}{7}) + 2 πn

Combine all the solutions

x = arccos (\frac{8 k - 7}{7}) + 2 πn, x = - arccos (\frac{8 k - 7}{7}) + 2 πn, x = arccos (- \frac{8 k - 7}{7}) + 2 πn, x = - arccos (- \frac{8 k - 7}{7}) + 2 πn

2 cos^{2} (x) - 7 cos (x) = - 3

sin (x) + 2 sin (2 x) = 0

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

cosh (x) cosh (1) - sinh (x) sinh (1) = 2

cos^{2} (x) = \frac{3 ( 1 - sin ( x ) )}{2}

What is the general solution for cos^2(x)=((8k-7))/7 ?
The general solution for cos^2(x)=((8k-7))/7 is