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Popular Trigonometry >

(sec^2(x))/2 =2cos^2(x)

Solution

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

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

Subtract

2 cos^{2} (x)

from both sides

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

Simplify

\frac{sec ^{2} ( x )}{2} - 2 cos^{2} (x) : \frac{sec ^{2} ( x ) - 4 cos ^{2} ( x )}{2}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{sec ^{2} ( x ) - 4 cos ^{2} ( x )}{2}

\frac{sec ^{2} ( x ) - 4 cos ^{2} ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sec^{2} (x) - 4 cos^{2} (x) = 0

Factor

sec^{2} (x) - 4 cos^{2} (x) : (sec (x) + 2 cos (x)) (sec (x) - 2 cos (x))

sec^{2} (x) - 4 cos^{2} (x)

Rewrite

sec^{2} (x) - 4 cos^{2} (x)

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

sec^{2} (x) - 4 cos^{2} (x)

Rewrite

4

2^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sec (x) + 2 cos (x)) (sec (x) - 2 cos (x))

(sec (x) + 2 cos (x)) (sec (x) - 2 cos (x)) = 0

Solving each part separately

sec (x) + 2 cos (x) = 0 or sec (x) - 2 cos (x) = 0

sec (x) + 2 cos (x) = 0 :

No Solution

sec (x) + 2 cos (x) = 0

Rewrite using trig identities

sec (x) + 2 cos (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

= sec (x) + \frac{2}{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 i, u = - 2 i

\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 i, u = - 2 i

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

For

x^{2} = f (a)

the solutions are

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

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

u = 2 i, u = - 2 i

Substitute back

u = sec (x)

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

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

sec (x) = 2 i :

No Solution

sec (x) = 2 i

No Solution

sec (x) = - 2 i :

No Solution

sec (x) = - 2 i

No Solution

Combine all the solutions

No Solution

sec (x) - 2 cos (x) = 0 : 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

sec (x) - 2 cos (x) = 0

Rewrite using trig identities

sec (x) - 2 cos (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Add

2

to both sides

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

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

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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