How to solve the equation cos^2x=1/2 using the formula of difference of two squares.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to solve (cos x)^2 = 1/2 using difference of squares.

(cos x)^2 = 1/2

=> (cos x)^2 - 1/2 = 0

a^2 - b^2 = (a - b)*(a + b)

=> (cos x - 1/sqrt 2)(cos x + 1/sqrt 2) = 0

cos x = 1/sqrt 2

=> x = arc cos (1/sqrt 2)

=> x = 45 degrees

cos x = -1/sqrt 2

=> x = arc cos (-1/sqrt 2)

=> x = 135 degrees

As the cosine function is periodic values of x at intervals of 360 degrees have the same cosine.

The required solution is 45 + n*360 degrees and 135 + n*360 degrees.

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll multiply by 2 both sides:

2(cos x)^2 = 1

We'll subtract 1 both sides to create the difference of 2 squares to the left side:

2(cos x)^2 - 1 = 0

We'll re-write the difference of squares as a product:

2(cos x)^2 - 1 = (sqrt2*cos x - 1)(sqrt2*cos x + 1)

We'll cancel out theproduct above:

(sqrt2*cos x - 1)(sqrt2*cos x + 1) = 0

We'll set each factor as zero:

sqrt2*cos x - 1 = 0

We'll add 1 both sides:

sqrt2*cos x = 1

We'll divide by sqrt 2:

cos x = 1/sqrt 2

cos x = sqrt 2/2

x = +arccos (sqrt 2/2) + 2k*pi

x = pi/4 + 2k*pi

x = - pi/4 + 2k*pi

We'll set the next factor as zero:

sqrt2*cos x + 1 = 0

We'll subtract 1 both sides:

sqrt2*cos x = -1

cos x = -1/sqrt 2

x = pi - arccos (sqrt 2/2) + 2k*pi

x = pi - pi/4 + 2k*pi

x = 3pi/4

The solutions of the equation are: {pi/4 + 2k*pi}U{3pi/4 + 2k*pi}.

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