What is the general solution for 4tan(x)=2sec^2(x) ?

The general solution for 4tan(x)=2sec^2(x) is x= pi/4+pin

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4tan(x)=2sec^2(x)

Solution

4 tan (x) = 2 sec^{2} (x)

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

4 tan (x) = 2 sec^{2} (x)

Subtract

2 sec^{2} (x)

from both sides

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

Rewrite using trig identities

- 2 sec^{2} (x) + 4 tan (x)

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

= - 2 (tan^{2} (x) + 1) + 4 tan (x)

- (1 + tan^{2} (x)) \cdot 2 + 4 tan (x) = 0

Solve by substitution

- (1 + tan^{2} (x)) \cdot 2 + 4 tan (x) = 0

Let:

tan (x) = u

- (1 + u^{2}) \cdot 2 + 4 u = 0

- (1 + u^{2}) \cdot 2 + 4 u = 0 : u = 1

- (1 + u^{2}) \cdot 2 + 4 u = 0

Expand

- (1 + u^{2}) \cdot 2 + 4 u : - 2 - 2 u^{2} + 4 u

- (1 + u^{2}) \cdot 2 + 4 u

= - 2 (1 + u^{2}) + 4 u

Expand

- 2 (1 + u^{2}) : - 2 - 2 u^{2}

- 2 (1 + u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 + (- 2) u^{2}

Apply minus-plus rules

+ (- a) = - a

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

Multiply the numbers:

2 \cdot 1 = 2

= - 2 - 2 u^{2}

= - 2 - 2 u^{2} + 4 u

- 2 - 2 u^{2} + 4 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- 2 u^{2} + 4 u - 2 = 0

Solve with the quadratic formula

- 2 u^{2} + 4 u - 2 = 0

Quadratic Equation Formula:

For

a = - 2, b = 4, c = - 2

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 2 ) ( - 2 )}{2 ( - 2 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 2 ) ( - 2 )}{2 ( - 2 )}

4^{2} - 4 (- 2) (- 2) = 0

4^{2} - 4 (- 2) (- 2)

Apply rule

- (- a) = a

= 4^{2} - 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 ( - 2 )}

u = \frac{- 4}{2 ( - 2 )}

\frac{- 4}{2 ( - 2 )} = 1

\frac{- 4}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 4}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

The solution to the quadratic equation is:

u = 1

Substitute back

u = tan (x)

tan (x) = 1

tan (x) = 1

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan(x)=2sec^2(x) ?
The general solution for 4tan(x)=2sec^2(x) is x= pi/4+pin