What is the general solution for sec^2(θ)=1-tan(θ) ?

The general solution for sec^2(θ)=1-tan(θ) is θ=(3pi)/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec^2(θ)=1-tan(θ)

Solution

sec^{2} (θ) = 1 - tan (θ)

Solution

θ = \frac{3 π}{4} + πn

Degrees

θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sec^{2} (θ) = 1 - tan (θ)

Subtract

1 - tan (θ)

from both sides

sec^{2} (θ) - 1 + tan (θ) = 0

Rewrite using trig identities

- 1 + sec^{2} (θ) + tan (θ)

Use the Pythagorean identity:

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

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

= tan (θ) + tan^{2} (θ)

Use the basic trigonometric identity:

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

= \frac{1}{cot ( θ )} + (\frac{1}{cot ( θ )})^{2}

(\frac{1}{cot ( θ )})^{2} = \frac{1}{cot ^{2} ( θ )}

(\frac{1}{cot ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cot ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( θ )}

= \frac{1}{cot ( θ )} + \frac{1}{cot ^{2} ( θ )}

\frac{1}{cot ^{2} ( θ )} + \frac{1}{cot ( θ )} = 0

Solve by substitution

\frac{1}{cot ^{2} ( θ )} + \frac{1}{cot ( θ )} = 0

Let:

cot (θ) = u

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

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

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

Multiply by LCM

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

Find Least Common Multiplier of

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

u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2}

u

= u^{2}

Multiply by LCM=

u^{2}

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

u

= u

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

1 + u = 0

1 + u = 0

1 + u = 0

Move

1

to the right side

1 + u = 0

Subtract

1

from both sides

1 + u - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1

Substitute back

u = cot (θ)

cot (θ) = - 1

cot (θ) = - 1

cot (θ) = - 1 : θ = \frac{3 π}{4} + πn

cot (θ) = - 1

General solutions for

cot (θ) = - 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{3 π}{4} + πn

θ = \frac{3 π}{4} + πn

Combine all the solutions

θ = \frac{3 π}{4} + πn

Popular Examples

cos^2(x)=-0.267

cos^{2} (x) = - 0.267

sin(x-4)=cos(9x+4)

sin (x - 4) = cos (9 x + 4)

solvefor x,0=-2sin(x)-4cos(2x)solve for

x, 0 = - 2 sin (x) - 4 cos (2 x)

tan(a)= 4/4

tan (a) = \frac{4}{4}

cos^2(θ)=2cos(θ)

cos^{2} (θ) = 2 cos (θ)

Frequently Asked Questions (FAQ)

What is the general solution for sec^2(θ)=1-tan(θ) ?
The general solution for sec^2(θ)=1-tan(θ) is θ=(3pi)/4+pin