multiple angle and product sum formula please help, test tom

[Dr.Fredrick]

Directions, If possible, find the exact solutions algebraically. in the period [0-2pi]
tan 2x- cotx=0
if you need the product to sum formulas and stuff just tell me and i can post them, or just tell me the steps. any help is appreciated, thanks

 

[BigGlenntheHeavy]

Dr. Fredrick, I misread an identity, sorry.

Here is the correct problem and answers.

$\displaystyle tan(2x)-cot(x) \ = \ 0$

$\displaystyle \frac{sin(2x)}{cos(2x)}-\frac{cos(x)}{sin(x)} \ = \ 0$

$\displaystyle \frac{2sin(x)cos(x)}{2cos^{2}(x)-1}-\frac{cos(x)}{sin(x)} \ = \ 0$

$\displaystyle \frac{2sin^{2}(x)cos(x)-cos(x)[2cos^{2}(x)-1]}{[2cos^{2}(x)-1]sin(x)]} \ = \ 0$

$\displaystyle \frac{2[1-cos^{2}(x)]cos(x)-2cos^{3}(x)+cos(x)}{[2cos^{2}(x)-1]sin(x)} \ = \ 0$

$\displaystyle \frac{2cos(x)-2cos^{3}(x)-2cos^{3}(x)+cos(x)}{[2cos^{2}(x)-1]sin(x)} \ = \ 0$

$\displaystyle \frac{3cos(x)-4cos^{3}(x)}{[2cos^{2}(x)-1]sin(x)} \ = \ 0$

$\displaystyle \frac{cos(x)[3-4cos^{2}(x)]}{[2cos^{2}(x)-1]sin(x)} \ = \ 0$

$\displaystyle cos(x) \ = \ 0, \ \implies \ x \ = \ \frac{\pi}{2}, \ \frac{3 \pi}{2}, \ cos^{2}(x) \ = \ \frac{3}{4}, \ \implies \ x \ = \ \frac{\pi}{6}, \ \frac{5 \pi}{6}, \ \frac{7 \pi}{6}, \ \frac{11 \pi}{6}$

$\displaystyle Hence, \ all \ are \ solutions, \ again \ I'm \ sorry.$

 

[Deleted member 4993]

Directions, If possible, find the exact solutions algebraically. in the period [0-2pi]
tan 2x- cotx=0
if you need the product to sum formulas and stuff just tell me and i can post them, or just tell me the steps. any help is appreciated, thanks

tan(2x) = cot(x)

sin(2x) * sin(x) = cos(2x)*cos(x)

cos(2x)*cos(x) - sin(2x) * sin(x) = 0

cos(3x) = 0

cos(3x) = cos[(2n+1)?/2]..............n = 0,1,2,3,4 & 5

x = (2n+1)?/6

x = ?/6, ?/2, 5?/6, 7?/6, 3?/2, 11?/6

 

[BigGlenntheHeavy]

Good show, Subhotosh Khan, as always there are more than one way to skin a cat.

How did you derive cos(3x) = cos[(2n+1)?/2]..............n=0,1,2,3,4,5 as I am curious?

 

[BigGlenntheHeavy]

Never mind, I got it.