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

The general solution for csc(x)cos^2(x)+sin^2(x)=csc(x) is x= pi/2+2pin

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Popular Trigonometry >

csc(x)cos^2(x)+sin^2(x)=csc(x)

Solution

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

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

csc (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

csc (x) (1 - sin^{2} (x)) : csc (x) - sin^{2} (x) csc (x)

csc (x) (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = csc (x), b = 1, c = sin^{2} (x)

= csc (x) \cdot 1 - csc (x) sin^{2} (x)

= 1 \cdot csc (x) - sin^{2} (x) csc (x)

Multiply:

1 \cdot csc (x) = csc (x)

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

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

Add similar elements:

- csc (x) + csc (x) = 0

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

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

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

Factor

sin^{2} (x) - csc (x) sin^{2} (x) : - sin^{2} (x) (csc (x) - 1)

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

Factor out common term

- sin^{2} (x)

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

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

Solving each part separately

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

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

csc (x) - 1 = 0

Move

1

to the right side

csc (x) - 1 = 0

Add

1

to both sides

csc (x) - 1 + 1 = 0 + 1

Simplify

csc (x) = 1

csc (x) = 1

General solutions for

csc (x) = 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{π}{2} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)cos^2(x)+sin^2(x)=csc(x) ?
The general solution for csc(x)cos^2(x)+sin^2(x)=csc(x) is x= pi/2+2pin