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

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

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

Use the Pythagorean identity:

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

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

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

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

Solve by substitution

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

Let:

sec (x) = u

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0 : u = 1, u = - 1

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

Multiply by LCM

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}} = 0

Find Least Common Multiplier of

u, u^{2} : u^{2}

u, u^{2}

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u

u^{2}

= u^{2}

Multiply by LCM=

u^{2}

- 1 \cdot u^{2} + \frac{- 1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

- 1 \cdot u^{2} + \frac{- 1 + u ^{2}}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

- 1 \cdot u^{2} : - u^{2}

- 1 \cdot u^{2}

Multiply:

1 \cdot u^{2} = u^{2}

= - u^{2}

Simplify

\frac{- 1 + u ^{2}}{u} u^{2} : u (u^{2} - 1)

\frac{- 1 + u ^{2}}{u} u^{2}

Multiply fractions:

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

= \frac{( - 1 + u ^{2} ) u ^{2}}{u}

Cancel the common factor:

u

= u (u^{2} - 1)

Simplify

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiply fractions:

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

= \frac{1 \cdot u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= 1

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- u^{2} + u (u^{2} - 1) + 1 = 0

- u^{2} + u (u^{2} - 1) + 1 = 0

- u^{2} + u (u^{2} - 1) + 1 = 0

Solve

- u^{2} + u (u^{2} - 1) + 1 = 0 : u = 1, u = - 1

- u^{2} + u (u^{2} - 1) + 1 = 0

Expand

- u^{2} + u (u^{2} - 1) + 1 : - u^{2} + u^{3} - u + 1

- u^{2} + u (u^{2} - 1) + 1

Expand

u (u^{2} - 1) : u^{3} - u

u (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = u, b = u^{2}, c = 1

= u u^{2} - u \cdot 1

= u^{2} u - 1 \cdot u

Simplify

u^{2} u - 1 \cdot u : u^{3} - u

u^{2} u - 1 \cdot u

u^{2} u = u^{3}

u^{2} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

1 \cdot u = u

1 \cdot u

Multiply:

1 \cdot u = u

= u

= u^{3} - u

= u^{3} - u

= - u^{2} + u^{3} - u + 1

- u^{2} + u^{3} - u + 1 = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{3} - u^{2} - u + 1 = 0

Factor

u^{3} - u^{2} - u + 1 : (u - 1)^{2} (u + 1)

u^{3} - u^{2} - u + 1

= (u^{3} - u^{2}) + (- u + 1)

Factor out

- 1

from

- u + 1 : - (u - 1)

- u + 1

Factor out common term

- 1

= - (u - 1)

Factor out

u^{2}

from

u^{3} - u^{2} : u^{2} (u - 1)

u^{3} - u^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= u u^{2} - u^{2}

Factor out common term

u^{2}

= u^{2} (u - 1)

= - (u - 1) + u^{2} (u - 1)

Factor out common term

u - 1

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

Factor

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u - 1) (u + 1) (u - 1)

Refine

= (u - 1)^{2} (u + 1)

(u - 1)^{2} (u + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 1 = 0 or u + 1 = 0

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

The solutions are

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 1 + \frac{- 1 + u ^{2}}{u} + \frac{1}{u ^{2}}

and compare to zero

u = 0

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

Substitute back

u = sec (x)

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

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

sec (x) = 1 : x = 2 πn

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

sec (x) = - 1 : x = π + 2 πn

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 = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

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