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

The general solution for tan(x)= 1/(cos^2(x)) is No Solution for x\in\mathbb{R}

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tan(x)= 1/(cos^2(x))

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

No Solution for x \in R

Solution steps

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

Subtract

\frac{1}{cos ^{2} ( x )}

from both sides

tan (x) - \frac{1}{cos ^{2} ( x )} = 0

Simplify

tan (x) - \frac{1}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) tan ( x ) - 1}{cos ^{2} ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

\frac{cos ^{2} ( x ) tan ( x ) - 1}{cos ^{2} ( x )} = 0

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

cos^{2} (x) tan (x) - 1 = 0

Rewrite using trig identities

- 1 + cos^{2} (x) tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x) cos (x)

= - 1 + sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

= - 1 + \frac{sin ( 2 x )}{2}

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

\frac{sin ( 2 x )}{2} = 1

\frac{sin ( 2 x )}{2} = 1

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = 1

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

arccsc (2 x) = \frac{π}{6}

2 csc (θ) = 2 cot (θ) + 3 csc (θ)

sin (\frac{1}{2} x) = - 1

9 cos^{2} (x) tan (x) = 4 tan (x)

3 tan (x) - 3 = 0, 0 \leq x \leq 2 π

What is the general solution for tan(x)= 1/(cos^2(x)) ?
The general solution for tan(x)= 1/(cos^2(x)) is No Solution for x\in\mathbb{R}