What is the general solution for cos(x+pi/4)+cos(x-pi/4)=1 ?

The general solution for cos(x+pi/4)+cos(x-pi/4)=1 is x= pi/4+2pin,x=(7pi)/4+2pin

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cos(x+pi/4)+cos(x-pi/4)=1

Solution

cos (x + \frac{π}{4}) + cos (x - \frac{π}{4}) = 1

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

cos (x + \frac{π}{4}) + cos (x - \frac{π}{4}) = 1

Rewrite using trig identities

cos (x + \frac{π}{4}) + cos (x - \frac{π}{4})

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2})

Simplify

2 cos (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2}) : 2 cos (x)

2 cos (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2})

\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2} = x

\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}

x + \frac{π}{4} + x - \frac{π}{4} = 2 x

x + \frac{π}{4} + x - \frac{π}{4}

Group like terms

= x + x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

x + x = 2 x

= 2 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 cos (x) cos (\frac{x - ( x - \frac{π}{4} ) + \frac{π}{4}}{2})

\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2} = \frac{π}{4}

\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2}

Join

x + \frac{π}{4} - (x - \frac{π}{4}) : \frac{π}{2}

x + \frac{π}{4} - (x - \frac{π}{4})

Convert element to fraction:

x = \frac{x 4}{4}, (x - \frac{π}{4}) = \frac{( x - \frac{π}{4} ) 4}{4}

= \frac{x \cdot 4}{4} + \frac{π}{4} - \frac{( x - \frac{π}{4} ) \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 4 + π - ( x - \frac{π}{4} ) \cdot 4}{4}

Expand

x \cdot 4 + π - (x - \frac{π}{4}) \cdot 4 : 2 π

x \cdot 4 + π - (x - \frac{π}{4}) \cdot 4

= 4 x + π - 4 (x - \frac{π}{4})

Expand

- 4 (x - \frac{π}{4}) : - 4 x + π

- 4 (x - \frac{π}{4})

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = x, c = \frac{π}{4}

= - 4 x - (- 4) \frac{π}{4}

Apply minus-plus rules

- (- a) = a

= - 4 x + 4 \cdot \frac{π}{4}

4 \cdot \frac{π}{4} = π

4 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 4}{4}

Cancel the common factor:

4

= π

= - 4 x + π

= x \cdot 4 + π - 4 x + π

Simplify

x \cdot 4 + π - 4 x + π : 2 π

x \cdot 4 + π - 4 x + π

Group like terms

= 4 x - 4 x + π + π

Add similar elements:

4 x - 4 x = 0

= π + π

Add similar elements:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= 2 cos (\frac{π}{4}) cos (x)

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 cos (x)

= 2 cos (x)

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 cos ( x )}{2} : cos (x)

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

Cancel the common factor:

2

= cos (x)

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

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

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

General solutions for

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

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 = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+pi/4)+cos(x-pi/4)=1 ?
The general solution for cos(x+pi/4)+cos(x-pi/4)=1 is x= pi/4+2pin,x=(7pi)/4+2pin