What is the general solution for 3/(cos^2(x))=3+2tan^2(x) ?

The general solution for 3/(cos^2(x))=3+2tan^2(x) is x=2pin,x=pi+2pin

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3/(cos^2(x))=3+2tan^2(x)

Solution

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

Subtract

3 + 2 tan^{2} (x)

from both sides

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x) = 0

Simplify

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x) : \frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x)

Convert element to fraction:

3 = \frac{3 cos ^{2} ( x )}{cos ^{2} ( x )}, 2 tan^{2} (x) = \frac{2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

\frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

3 - 3 cos^{2} (x) - 2 tan^{2} (x) cos^{2} (x) = 0

Express with sin, cos

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x) = 0

Simplify

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

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

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= sin^{2} (x) \cdot 2

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

3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0

Add

2 sin^{2} (x)

to both sides

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

Square both sides

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

Subtract

(2 sin^{2} (x))^{2}

from both sides

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x) = 0

Factor

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x) : (3 - 3 cos^{2} (x) + 2 sin^{2} (x)) (3 - 3 cos^{2} (x) - 2 sin^{2} (x))

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

Rewrite

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

(3 - 3 cos^{2} (x))^{2} - (2 sin^{2} (x))^{2}

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

Rewrite

4

2^{2}

= (3 - 3 cos^{2} (x))^{2} - 2^{2} sin^{4} (x)

Apply exponent rule:

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

sin^{4} (x) = (sin^{2} (x))^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

(3 - 3 cos^{2} (x))^{2} - (2 sin^{2} (x))^{2} = ((3 - 3 cos^{2} (x)) + 2 sin^{2} (x)) ((3 - 3 cos^{2} (x)) - 2 sin^{2} (x))

= ((3 - 3 cos^{2} (x)) + 2 sin^{2} (x)) ((3 - 3 cos^{2} (x)) - 2 sin^{2} (x))

Refine

= (2 sin^{2} (x) - 3 cos^{2} (x) + 3) (- 3 cos^{2} (x) - 2 sin^{2} (x) + 3)

(3 - 3 cos^{2} (x) + 2 sin^{2} (x)) (3 - 3 cos^{2} (x) - 2 sin^{2} (x)) = 0

Solving each part separately

3 - 3 cos^{2} (x) + 2 sin^{2} (x) = 0 or 3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0

3 - 3 cos^{2} (x) + 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

3 - 3 cos^{2} (x) + 2 sin^{2} (x) = 0

Rewrite using trig identities

3 + 2 sin^{2} (x) - 3 cos^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

= 3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Simplify

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x)) : 5 sin^{2} (x)

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Expand

- 3 (1 - sin^{2} (x)) : - 3 + 3 sin^{2} (x)

- 3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

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

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= - 3 + 3 sin^{2} (x)

= 3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

Simplify

3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x) : 5 sin^{2} (x)

3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

Group like terms

= 2 sin^{2} (x) + 3 sin^{2} (x) + 3 - 3

Add similar elements:

2 sin^{2} (x) + 3 sin^{2} (x) = 5 sin^{2} (x)

= 5 sin^{2} (x) + 3 - 3

3 - 3 = 0

= 5 sin^{2} (x)

= 5 sin^{2} (x)

= 5 sin^{2} (x)

5 sin^{2} (x) = 0

Divide both sides by

5

5 sin^{2} (x) = 0

Divide both sides by

5

5 sin^{2} (x) = 0

Divide both sides by

5

\frac{5 sin ^{2} ( x )}{5} = \frac{0}{5}

Simplify

sin^{2} (x) = 0

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

3 - 3 cos^{2} (x) - 2 sin^{2} (x) = 0

Rewrite using trig identities

3 - 2 sin^{2} (x) - 3 cos^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

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

Simplify

3 - 2 sin^{2} (x) - 3 (1 - sin^{2} (x)) : sin^{2} (x)

3 - 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

Expand

- 3 (1 - sin^{2} (x)) : - 3 + 3 sin^{2} (x)

- 3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

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

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= - 3 + 3 sin^{2} (x)

= 3 - 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

Simplify

3 - 2 sin^{2} (x) - 3 + 3 sin^{2} (x) : sin^{2} (x)

3 - 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

Group like terms

= - 2 sin^{2} (x) + 3 sin^{2} (x) + 3 - 3

Add similar elements:

- 2 sin^{2} (x) + 3 sin^{2} (x) = sin^{2} (x)

= sin^{2} (x) + 3 - 3

3 - 3 = 0

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

Remove the ones that don't agree with the equation.

Check the solution

2 πn :

True

2 πn

Plug in

n = 1

2 π 1

For

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

plug in

x = 2 π 1

\frac{3}{cos ^{2} ( 2 π 1 )} = 3 + 2 tan^{2} (2 π 1)

Refine

3 = 3

\Rightarrow True

Check the solution

π + 2 πn :

True

π + 2 πn

Plug in

n = 1

π + 2 π 1

For

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

plug in

x = π + 2 π 1

\frac{3}{cos ^{2} ( π + 2 π 1 )} = 3 + 2 tan^{2} (π + 2 π 1)

Refine

3 = 3

\Rightarrow True

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3/(cos^2(x))=3+2tan^2(x) ?
The general solution for 3/(cos^2(x))=3+2tan^2(x) is x=2pin,x=pi+2pin