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

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

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Popular >

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

tan^{2} (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Factor

sec^{2} (x) - tan^{2} (x) cos^{2} (x) : (sec (x) + tan (x) cos (x)) (sec (x) - tan (x) cos (x))

sec^{2} (x) - tan^{2} (x) cos^{2} (x)

Rewrite

tan^{2} (x) cos^{2} (x)

(tan (x) cos (x))^{2}

tan^{2} (x) cos^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

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

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

= (sec (x) + tan (x) cos (x)) (sec (x) - tan (x) cos (x))

(sec (x) + tan (x) cos (x)) (sec (x) - tan (x) cos (x)) = 0

Solving each part separately

sec (x) + tan (x) cos (x) = 0 or sec (x) - tan (x) cos (x) = 0

sec (x) + tan (x) cos (x) = 0 :

No Solution

sec (x) + tan (x) cos (x) = 0

Express with sin, cos

sec (x) + cos (x) tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} + cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

\frac{1}{cos ( x )} + cos (x) \frac{sin ( x )}{cos ( x )} : \frac{1 + sin ( x ) cos ( x )}{cos ( x )}

\frac{1}{cos ( x )} + cos (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

1 + cos (x) sin (x) = 0

Rewrite using trig identities

1 + cos (x) sin (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

Subtract

1

from 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} = 2 (- 1)

Simplify

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

sec (x) - tan (x) cos (x) = 0 :

No Solution

sec (x) - tan (x) cos (x) = 0

Express with sin, cos

sec (x) - cos (x) tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Use the basic trigonometric identity:

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

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

Simplify

\frac{1}{cos ( x )} - cos (x) \frac{sin ( x )}{cos ( x )} : \frac{1 - sin ( x ) cos ( x )}{cos ( x )}

\frac{1}{cos ( x )} - cos (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

1 - cos (x) sin (x) = 0

Rewrite using trig identities

1 - cos (x) sin (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

Subtract

1

from 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

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

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

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

2sin^2(x/3)=1,-pi<= ,x<= pi

2 sin^{2} (\frac{x}{3}) = 1, - π \leq, x \leq π

cos^2(x)+sin^2(x)=sin(x)+1/2

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

sin(A)=0.53

sin (A) = 0.53

tan(a)=sqrt(3),cos(a)

tan (a) = 3, cos (a)

Frequently Asked Questions (FAQ)

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