What is the general solution for 2sec(x)cos^2(x)=csc^2(x)-cot^2(x) ?

The general solution for 2sec(x)cos^2(x)=csc^2(x)-cot^2(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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2sec(x)cos^2(x)=csc^2(x)-cot^2(x)

Solution

2 sec (x) cos^{2} (x) = csc^{2} (x) - cot^{2} (x)

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sec (x) cos^{2} (x) = csc^{2} (x) - cot^{2} (x)

Subtract

csc^{2} (x) - cot^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

sec (x)

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

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

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

csc^{2} (x) - cot^{2} (x) = 1

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

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

Multiply both sides by

sec (x)

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

Multiply both sides by

sec (x)

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

Simplify

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

Simplify

\frac{2}{sec ( x )} sec (x) : 2

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

Multiply fractions:

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

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

Cancel the common factor:

sec (x)

= 2

Simplify

- 1 \cdot sec (x) : - sec (x)

- 1 \cdot sec (x)

Multiply:

1 \cdot sec (x) = sec (x)

= - sec (x)

Simplify

0 \cdot sec (x) : 0

0 \cdot sec (x)

Apply rule

0 \cdot a = 0

= 0

2 - sec (x) = 0

2 - sec (x) = 0

2 - sec (x) = 0

Move

2

to the right side

2 - sec (x) = 0

Subtract

2

from both sides

2 - sec (x) - 2 = 0 - 2

Simplify

- sec (x) = - 2

- sec (x) = - 2

Divide both sides by

- 1

- sec (x) = - 2

Divide both sides by

- 1

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

Simplify

sec (x) = 2

sec (x) = 2

Verify Solutions

Find undefined (singularity) points:

sec (x) = 0

Take the denominator(s) of

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

and compare to zero

sec (x) = 0

The following points are undefined

sec (x) = 0

Combine undefined points with solutions:

sec (x) = 2

General solutions for

sec (x) = 2

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Popular Examples

1=7cos(pi/3 t)

1 = 7 cos (\frac{π}{3} t)

12cos^2(x)+2cos(x)-2=0

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

sin(θ)=-0.94

sin (θ) = - 0.94

sin(θ)=0,0<= θ<= 2pi

sin (θ) = 0, 0 \leq θ \leq 2 π

sin(θ)=-0.71

sin (θ) = - 0.71

Frequently Asked Questions (FAQ)

What is the general solution for 2sec(x)cos^2(x)=csc^2(x)-cot^2(x) ?
The general solution for 2sec(x)cos^2(x)=csc^2(x)-cot^2(x) is x= pi/3+2pin,x=(5pi)/3+2pin