What is the general solution for (sec^2(-1))/(tan(x))=tan(x) ?

The general solution for (sec^2(-1))/(tan(x))=tan(x) is x=1.07542…+pin,x=-1.07542…+pin

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(sec^2(-1))/(tan(x))=tan(x)

Solution

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

Solution

x = 1.07542 \dots + πn, x = - 1.07542 \dots + πn

Degrees

x = 61.61754 \dots^{\circ} + 18 0^{\circ} n, x = - 61.61754 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

sec (- 1) = sec (1)

sec (- 1)

Use the following property:

sec (- x) = sec (x)

sec (- 1) = sec (1)

= sec (1)

\frac{sec ^{2} ( 1 )}{tan ( x )} = tan (x)

Solve by substitution

\frac{sec ^{2} ( 1 )}{tan ( x )} = tan (x)

Let:

tan (x) = u

\frac{sec ^{2} ( 1 )}{u} = u

\frac{sec ^{2} ( 1 )}{u} = u : u = sec (1), u = - sec (1)

\frac{sec ^{2} ( 1 )}{u} = u

Multiply both sides by

u

\frac{sec ^{2} ( 1 )}{u} = u

Multiply both sides by

u

\frac{sec ^{2} ( 1 )}{u} u = uu

Simplify

uu : u^{2}

\frac{sec ^{2} ( 1 )}{u} u = 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}

sec^{2} (1) = u^{2}

sec^{2} (1) = u^{2}

Solve

sec^{2} (1) = u^{2} : u = sec (1), u = - sec (1)

sec^{2} (1) = u^{2}

Switch sides

u^{2} = sec^{2} (1)

For

x^{2} = f (a)

the solutions are

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

u = sec^{2} (1), u = - sec^{2} (1)

sec^{2} (1) = sec (1)

sec^{2} (1)

Apply radical rule:

assuming

a \geq 0

= sec (1)

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

- sec^{2} (1)

Apply radical rule:

assuming

a \geq 0

= - sec (1)

u = sec (1), u = - sec (1)

u = sec (1), u = - sec (1)

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{sec ^{2} ( 1 )}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = sec (1), u = - sec (1)

Substitute back

u = tan (x)

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

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

tan (x) = sec (1) : x = arctan (sec (1)) + πn

tan (x) = sec (1)

Apply trig inverse properties

tan (x) = sec (1)

General solutions for

tan (x) = sec (1)

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (sec (1)) + πn

x = arctan (sec (1)) + πn

tan (x) = - sec (1) : x = arctan (- sec (1)) + πn

tan (x) = - sec (1)

Apply trig inverse properties

tan (x) = - sec (1)

General solutions for

tan (x) = - sec (1)

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- sec (1)) + πn

x = arctan (- sec (1)) + πn

Combine all the solutions

x = arctan (sec (1)) + πn, x = arctan (- sec (1)) + πn

Show solutions in decimal form

x = 1.07542 \dots + πn, x = - 1.07542 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sec^2(-1))/(tan(x))=tan(x) ?
The general solution for (sec^2(-1))/(tan(x))=tan(x) is x=1.07542…+pin,x=-1.07542…+pin