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

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

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sec^2(x)tan^2(x)+3sec^2(x)-2tan^2(x)=3

Solution

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

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

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

Subtract

3

from both sides

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

Rewrite using trig identities

- 3 - 2 tan^{2} (x) + 3 sec^{2} (x) + sec^{2} (x) tan^{2} (x)

Use the Pythagorean identity:

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

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

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

Simplify

- 3 - 2 (sec^{2} (x) - 1) + 3 sec^{2} (x) + sec^{2} (x) (sec^{2} (x) - 1) : sec^{4} (x) - 1

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Expand

sec^{2} (x) (sec^{2} (x) - 1) : sec^{4} (x) - sec^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

sec^{2} (x) sec^{2} (x) - 1 \cdot sec^{2} (x) : sec^{4} (x) - sec^{2} (x)

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

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

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

Apply exponent rule:

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

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

= sec^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sec^{4} (x)

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

1 \cdot sec^{2} (x)

Multiply:

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

= sec^{2} (x)

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

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

= - 3 - 2 sec^{2} (x) + 2 + 3 sec^{2} (x) + sec^{4} (x) - sec^{2} (x)

Simplify

- 3 - 2 sec^{2} (x) + 2 + 3 sec^{2} (x) + sec^{4} (x) - sec^{2} (x) : sec^{4} (x) - 1

- 3 - 2 sec^{2} (x) + 2 + 3 sec^{2} (x) + sec^{4} (x) - sec^{2} (x)

Group like terms

= - 2 sec^{2} (x) + 3 sec^{2} (x) + sec^{4} (x) - sec^{2} (x) - 3 + 2

Add similar elements:

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

= sec^{4} (x) - 3 + 2

Add/Subtract the numbers:

- 3 + 2 = - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

- 1 + sec^{4} (x) = 0

Solve by substitution

- 1 + sec^{4} (x) = 0

Let:

sec (x) = u

- 1 + u^{4} = 0

- 1 + u^{4} = 0 : u = 1, u = - 1, u = i, u = - i

- 1 + u^{4} = 0

Move

1

to the right side

- 1 + u^{4} = 0

Add

1

to both sides

- 1 + u^{4} + 1 = 0 + 1

Simplify

u^{4} = 1

u^{4} = 1

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = 1

Solve

v^{2} = 1 : v = 1, v = - 1

v^{2} = 1

For

(g (x))^{2} = f (a)

the solutions are

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

v = 1, v = - 1

v = 1, v = - 1

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

Apply rule

1 = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Solve

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

Apply rule

1 = 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

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

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

The solutions are

u = 1, u = - 1, u = i, u = - i

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

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 = 0 + 2 πn

x = 0 + 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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

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 = π + 2 πn

x = π + 2 πn

sec (x) = i :

No Solution

sec (x) = i

No Solution

sec (x) = - i :

No Solution

sec (x) = - i

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

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