What is the general solution for tan(θ/2)=csc^2(θ)-cot^2(θ) ?

The general solution for tan(θ/2)=csc^2(θ)-cot^2(θ) is θ= pi/2+2pin

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Popular >

tan(θ/2)=csc^2(θ)-cot^2(θ)

Solution

tan (\frac{θ}{2}) = csc^{2} (θ) - cot^{2} (θ)

Solution

θ = \frac{π}{2} + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (\frac{θ}{2}) = csc^{2} (θ) - cot^{2} (θ)

Subtract

csc^{2} (θ) - cot^{2} (θ)

from both sides

tan (\frac{θ}{2}) - csc^{2} (θ) + cot^{2} (θ) = 0

Rewrite using trig identities

tan (\frac{θ}{2}) - csc^{2} (θ) + cot^{2} (θ)

Use the Pythagorean identity:

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

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

= tan (\frac{θ}{2}) - 1

tan (\frac{θ}{2}) - 1 = 0

Move

1

to the right side

tan (\frac{θ}{2}) - 1 = 0

Add

1

to both sides

tan (\frac{θ}{2}) - 1 + 1 = 0 + 1

Simplify

tan (\frac{θ}{2}) = 1

tan (\frac{θ}{2}) = 1

General solutions for

tan (\frac{θ}{2}) = 1

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{θ}{2} = \frac{π}{4} + πn

\frac{θ}{2} = \frac{π}{4} + πn

Solve

\frac{θ}{2} = \frac{π}{4} + πn : θ = \frac{π}{2} + 2 πn

\frac{θ}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot \frac{π}{4} + 2 πn : \frac{π}{2} + 2 πn

2 \cdot \frac{π}{4} + 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(θ/2)=csc^2(θ)-cot^2(θ) ?
The general solution for tan(θ/2)=csc^2(θ)-cot^2(θ) is θ= pi/2+2pin