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

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

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(tan^2(x)(1+cot^2(x)))/(sec(x))=cos(x)

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

cos (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

cot (x) = \frac{1}{tan ( x )}

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

Simplify

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

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

Expand

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

tan^{2} (x) (1 + \frac{1}{tan ^{2} ( x )})

Apply the distributive law:

a (b + c) = ab + a c

a = tan^{2} (x), b = 1, c = \frac{1}{tan ^{2} ( x )}

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

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

Simplify

1 \cdot tan^{2} (x) + \frac{1}{tan ^{2} ( x )} tan^{2} (x) : tan^{2} (x) + 1

1 \cdot tan^{2} (x) + \frac{1}{tan ^{2} ( x )} tan^{2} (x)

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

1 \cdot tan^{2} (x)

Multiply:

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

= tan^{2} (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

tan^{2} (x)

= 1

= tan^{2} (x) + 1

= tan^{2} (x) + 1

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

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

sec (x) cos (x) = 1

sec (x) cos (x)

Express with sin, cos

sec (x) cos (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= 1

= 1

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

tan^{2} (x) = 0

tan^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

Since the equation is undefined for:

πn

No Solution for x \in R

Popular Examples

4cos^2(x)tan(x)-tan(x)=0

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

sin(2x)+sin^2(x)+cos^2(x)=0

sin (2 x) + sin^{2} (x) + cos^{2} (x) = 0

csc(x)= 1/(cos(x))

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

csc(θ)=(1.201)

csc (θ) = (1.201)

3cos(θ)+2=5sec(θ),0<= θ<2pi

3 cos (θ) + 2 = 5 sec (θ), 0 \leq θ < 2 π

Frequently Asked Questions (FAQ)

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