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Popular Trigonometry >

cos(x)=sec(x)(1-cos^2(x))

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

cos (x) - sec (x) (1 - cos^{2} (x)) = 0

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= \frac{1}{sec ( x )} - (1 - (\frac{1}{sec ( x )})^{2}) sec (x)

Simplify

\frac{1}{sec ( x )} - (1 - (\frac{1}{sec ( x )})^{2}) sec (x) : \frac{2}{sec ( x )} - sec (x)

\frac{1}{sec ( x )} - (1 - (\frac{1}{sec ( x )})^{2}) sec (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sec ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= \frac{1}{sec ( x )} - sec (x) (- \frac{1}{sec ^{2} ( x )} + 1)

= \frac{1}{sec ( x )} - sec (x) (1 - \frac{1}{sec ^{2} ( x )})

Expand

- sec (x) (1 - \frac{1}{sec ^{2} ( x )}) : - sec (x) + \frac{1}{sec ( x )}

- sec (x) (1 - \frac{1}{sec ^{2} ( x )})

Apply the distributive law:

a (b - c) = ab - a c

a = - sec (x), b = 1, c = \frac{1}{sec ^{2} ( x )}

= - sec (x) \cdot 1 - (- sec (x)) \frac{1}{sec ^{2} ( x )}

Apply minus-plus rules

- (- a) = a

= - 1 \cdot sec (x) + \frac{1}{sec ^{2} ( x )} sec (x)

Simplify

- 1 \cdot sec (x) + \frac{1}{sec ^{2} ( x )} sec (x) : - sec (x) + \frac{1}{sec ( x )}

- 1 \cdot sec (x) + \frac{1}{sec ^{2} ( x )} sec (x)

1 \cdot sec (x) = sec (x)

1 \cdot sec (x)

Multiply:

1 \cdot sec (x) = sec (x)

= sec (x)

\frac{1}{sec ^{2} ( x )} sec (x) = \frac{1}{sec ( x )}

\frac{1}{sec ^{2} ( x )} sec (x)

Multiply fractions:

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

= \frac{1 \cdot sec ( x )}{sec ^{2} ( x )}

Multiply:

1 \cdot sec (x) = sec (x)

= \frac{sec ( x )}{sec ^{2} ( x )}

Cancel the common factor:

sec (x)

= \frac{1}{sec ( x )}

= - sec (x) + \frac{1}{sec ( x )}

= - sec (x) + \frac{1}{sec ( x )}

= \frac{1}{sec ( x )} - sec (x) + \frac{1}{sec ( x )}

Combine the fractions

\frac{1}{sec ( x )} + \frac{1}{sec ( x )} : \frac{2}{sec ( x )}

Apply rule

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

= \frac{1 + 1}{sec ( x )}

Add the numbers:

1 + 1 = 2

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

= \frac{2}{sec ( x )} - sec (x)

= \frac{2}{sec ( x )} - sec (x)

\frac{2}{sec ( x )} - sec (x) = 0

Solve by substitution

\frac{2}{sec ( x )} - sec (x) = 0

Let:

sec (x) = u

\frac{2}{u} - u = 0

\frac{2}{u} - u = 0 : u = 2, u = - 2

\frac{2}{u} - u = 0

Multiply both sides by

u

\frac{2}{u} - u = 0

Multiply both sides by

u

\frac{2}{u} u - uu = 0 \cdot u

Simplify

\frac{2}{u} u - uu = 0 \cdot u

Simplify

\frac{2}{u} u : 2

\frac{2}{u} u

Multiply fractions:

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

= \frac{2 u}{u}

Cancel the common factor:

u

= 2

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

2 - u^{2} = 0

2 - u^{2} = 0

2 - u^{2} = 0

Solve

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

Move

2

to the right side

2 - u^{2} = 0

Subtract

2

from both sides

2 - u^{2} - 2 = 0 - 2

Simplify

- u^{2} = - 2

- u^{2} = - 2

Divide both sides by

- 1

- u^{2} = - 2

Divide both sides by

- 1

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

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

u = 2, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2}{u} - u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 2

Substitute back

u = sec (x)

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = 2

General solutions for

sec (x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = - 2

General solutions for

sec (x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Graph

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