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

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

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sin^2(x)tan^2(x)=tan^2(x)+sin^2(x)

Solution

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

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

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

Subtract

tan^{2} (x) + sin^{2} (x)

from both sides

sin^{2} (x) tan^{2} (x) - tan^{2} (x) - sin^{2} (x) = 0

Express with sin, cos

- sin^{2} (x) - tan^{2} (x) + sin^{2} (x) tan^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

- sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2} + sin^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} : \frac{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

Apply exponent rule:

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

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

sin^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

sin^{2} (x) sin^{2} (x) = sin^{4} (x)

sin^{2} (x) sin^{2} (x)

Apply exponent rule:

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

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

= sin^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sin^{4} (x)

= \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin^{2} (x) - \frac{sin ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

Combine the fractions

- \frac{sin ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{4} ( x )}{cos ^{2} ( x )} : \frac{- sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

Apply rule

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

= \frac{- sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin^{2} (x) + \frac{sin ^{4} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

Convert element to fraction:

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

= - \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{- sin ^{2} ( x ) + sin ^{4} ( 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{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

= \frac{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

\frac{- sin ^{2} ( x ) + sin ^{4} ( x ) - cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )} = 0

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

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

Factor

- sin^{2} (x) + sin^{4} (x) - cos^{2} (x) sin^{2} (x) : - sin^{2} (x) (1 - sin^{2} (x) + cos^{2} (x))

- sin^{2} (x) + sin^{4} (x) - cos^{2} (x) sin^{2} (x)

Apply exponent rule:

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

sin^{4} (x) = sin^{2} (x) sin^{2} (x)

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

Factor out common term

- sin^{2} (x)

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

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

Solving each part separately

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Apply rule

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

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

1 - sin^{2} (x) + cos^{2} (x) = 0 : x = \frac{π}{2} + πn

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= 1 + cos (2 x)

1 + cos (2 x) = 0

Move

1

to the right side

1 + cos (2 x) = 0

Subtract

1

from both sides

1 + cos (2 x) - 1 = 0 - 1

Simplify

cos (2 x) = - 1

cos (2 x) = - 1

General solutions for

cos (2 x) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

2 x = π + 2 πn

2 x = π + 2 πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + πn

Since the equation is undefined for:

\frac{π}{2} + πn

x = 2 πn, x = π + 2 πn

Popular Examples

solvefor θ,cot(θ)=-pi/6 solve for

θ, cot (θ) = - \frac{π}{6}

tan(x)+cos(x)=0

tan (x) + cos (x) = 0

tan(θ)=sin(2θ)

tan (θ) = sin (2 θ)

sin(θ_{1})=-10/13

sin (θ_{1}) = - \frac{10}{13}

sin(a^{-1})=(2.6)/(0.1733)

sin (a^{- 1}) = \frac{2.6}{0.1733}

Frequently Asked Questions (FAQ)

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