What is the general solution for 3(tan(x)+sin^2(x))-4=4tan(x)-3cos^2(x) ?

The general solution for 3(tan(x)+sin^2(x))-4=4tan(x)-3cos^2(x) is x=(3pi)/4+pin

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3(tan(x)+sin^2(x))-4=4tan(x)-3cos^2(x)

Solution

3 (tan (x) + sin^{2} (x)) - 4 = 4 tan (x) - 3 cos^{2} (x)

Solution

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

3 (tan (x) + sin^{2} (x)) - 4 = 4 tan (x) - 3 cos^{2} (x)

Subtract

4 tan (x) - 3 cos^{2} (x)

from both sides

- tan (x) + 3 sin^{2} (x) + 3 cos^{2} (x) - 4 = 0

Rewrite using trig identities

- 4 - tan (x) + 3 cos^{2} (x) + 3 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 4 - tan (x) + 3 (1 - sin^{2} (x)) + 3 sin^{2} (x)

Simplify

- 4 - tan (x) + 3 (1 - sin^{2} (x)) + 3 sin^{2} (x) : - tan (x) - 1

- 4 - tan (x) + 3 (1 - sin^{2} (x)) + 3 sin^{2} (x)

Expand

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 sin^{2} (x)

= - 4 - tan (x) + 3 - 3 sin^{2} (x) + 3 sin^{2} (x)

Simplify

- 4 - tan (x) + 3 - 3 sin^{2} (x) + 3 sin^{2} (x) : - tan (x) - 1

- 4 - tan (x) + 3 - 3 sin^{2} (x) + 3 sin^{2} (x)

Add similar elements:

- 3 sin^{2} (x) + 3 sin^{2} (x) = 0

= - 4 - tan (x) + 3

Group like terms

= - tan (x) - 4 + 3

Add/Subtract the numbers:

- 4 + 3 = - 1

= - tan (x) - 1

= - tan (x) - 1

= - tan (x) - 1

- 1 - tan (x) = 0

Move

1

to the right side

- 1 - tan (x) = 0

Add

1

to both sides

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

Simplify

- tan (x) = 1

- tan (x) = 1

Divide both sides by

- 1

- tan (x) = 1

Divide both sides by

- 1

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

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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\frac{1}{3} = tan (θ)

(sec^{(2)}(θ))(cos(θ))=2cos(θ)

(sec^{(2)} (θ)) (cos (θ)) = 2 cos (θ)

sin(x)*tan(x)+1=sin(x)+tan(x)

sin (x) \cdot tan (x) + 1 = sin (x) + tan (x)

1.33*sin(56)=1*sin(x)

1.33 \cdot sin (5 6^{\circ}) = 1 \cdot sin (x)

cosh(x)=sinh(x)

cosh (x) = sinh (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3(tan(x)+sin^2(x))-4=4tan(x)-3cos^2(x) ?
The general solution for 3(tan(x)+sin^2(x))-4=4tan(x)-3cos^2(x) is x=(3pi)/4+pin