csc(x)=cot(x)-1

Question

csc(x)=cot(x)-1

Answer 1

1/sin(x)=[cos(x)/sin(x)]-1

1=cos(x)-sin(x),,,,,,,,x=0+2πk not valid for answer since sin(0+2πk)=0 and 1/sin(x) is not valid

cos(x)-sin(x)=√2*cos(x+π/4),,,,,important identity, -π/4=tan^-1(-1/1)

x+π/4=cos^-1(√2/2)+2πk

and

x+π/4=-cos(√2/2)+2πk

from first equation, x=0 which is not available for valid answer

from second equation

x=-π/2+2πk,,,,k=0,+/-1,+/-2,...

Answer 2

Hi Joshua G.,

You should plot these on Desmos to solve graphically.

Algebraically, one way shown below. I'm sure there are others.

csc(x) = cot(x) - 1

1/sin(x) = cos(x) / sin(x) - 1

1/sin(x) = [cos(x) - sin(x)] / sin(x)

sin(x) = sin(x) / [cos(x) - sin(x)]

sin(x)[cos(x) - sin(x)] = sin(x)

sin(x)[cos(x) - six(x)] - sin(x) = 0

sin(x)[cos(x) - sin(x) - 1] = 0

cos(x) - sin(x) - 1 = 0 sin(x) = 0

cos(x) = sin(x) + 1 x = 0 + πk, csc(x) ≠ 0

cos²(x) = sin²(x) + 2sin(x) + 1

1 - sin²(x) = sin²(x) + 2sin(x) + 1

2sin²(x) + 2sin(x) = 0

2sin(x)[sin(x) + 1] = 0

sin(x) = -1 2sin(x) ≠ 0

x = 3π/2 + 2πk

I hope this helps, Joe.