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Popular >

csc^2(x)+cot^2(x)=2

Solution

csc^{2} (x) + cot^{2} (x) = 2

Solution

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

+1

Degrees

x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 125.26438 \dots^{\circ} + 36 0^{\circ} n, x = - 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 234.73561 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (x) + cot^{2} (x) = 2

Subtract

2

from both sides

csc^{2} (x) + cot^{2} (x) - 2 = 0

Rewrite using trig identities

- 2 + cot^{2} (x) + csc^{2} (x)

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 2 + csc^{2} (x) - 1 + csc^{2} (x)

Simplify

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

- 2 + csc^{2} (x) - 1 + csc^{2} (x)

Group like terms

= csc^{2} (x) + csc^{2} (x) - 2 - 1

Add similar elements:

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

= 2 csc^{2} (x) - 2 - 1

Subtract the numbers:

- 2 - 1 = - 3

= 2 csc^{2} (x) - 3

= 2 csc^{2} (x) - 3

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

Solve by substitution

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

Let:

csc (x) = u

- 3 + 2 u^{2} = 0

- 3 + 2 u^{2} = 0 : u = \frac{3}{2}, u = - \frac{3}{2}

- 3 + 2 u^{2} = 0

Move

3

to the right side

- 3 + 2 u^{2} = 0

Add

3

to both sides

- 3 + 2 u^{2} + 3 = 0 + 3

Simplify

2 u^{2} = 3

2 u^{2} = 3

Divide both sides by

2

2 u^{2} = 3

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{3}{2}

Simplify

u^{2} = \frac{3}{2}

u^{2} = \frac{3}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = csc (x)

csc (x) = \frac{3}{2}, csc (x) = - \frac{3}{2}

csc (x) = \frac{3}{2}, csc (x) = - \frac{3}{2}

csc (x) = \frac{3}{2} : x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

csc (x) = \frac{3}{2}

Apply trig inverse properties

csc (x) = \frac{3}{2}

General solutions for

csc (x) = \frac{3}{2}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

csc (x) = - \frac{3}{2} : x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

csc (x) = - \frac{3}{2}

Apply trig inverse properties

csc (x) = - \frac{3}{2}

General solutions for

csc (x) = - \frac{3}{2}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

Combine all the solutions

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn, x = arccsc (- \frac{3}{2}) + 2 πn, x = π + arccsc (\frac{3}{2}) + 2 πn

Show solutions in decimal form

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

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