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

The general solution for (1+tan^2(x))cos^2(x)=sec(x) is x=2pin

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

Solution

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

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

sec (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

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

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

Simplify

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

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

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

1 \cdot cos^{2} (x)

Multiply:

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

= cos^{2} (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= sin^{2} (x)

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

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

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

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

Use the Pythagorean identity:

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

= - sec (x) + 1

- sec (x) + 1 = 0

Move

1

to the right side

- sec (x) + 1 = 0

Subtract

1

from both sides

- sec (x) + 1 - 1 = 0 - 1

Simplify

- sec (x) = - 1

- sec (x) = - 1

Divide both sides by

- 1

- sec (x) = - 1

Divide both sides by

- 1

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

Simplify

sec (x) = 1

sec (x) = 1

General solutions for

sec (x) = 1

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Popular Examples

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

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

cos^6(x)=-cos^2(x)

cos^{6} (x) = - cos^{2} (x)

2sin^3(x)-5sin^2(x)+2sin(x)=0

2 sin^{3} (x) - 5 sin^{2} (x) + 2 sin (x) = 0

(cos^2(a)-3cos(a)+2)/(sin^2(a))=1

\frac{cos ^{2} ( a ) - 3 cos ( a ) + 2}{sin ^{2} ( a )} = 1

(sin(x)-(sqrt(2)))/2 =0

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

Frequently Asked Questions (FAQ)

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