1. cot x sec4x = cot x + 2 tan x + tan3x

2. (sin x) (tan x cos x - cot x cos x) = 1 - 2 cos2x

3. 1 + sec2x sin2x = sec2x

4. - tan2x + sec2x = 1

1. cot (x) sec⁴ (x) = cot (x) + 2tan (x) + tan (3x)

cot (x) sec⁴ (x) cot (x) sec⁴ (x)

0 = cos⁴ (x) + 2cos⁴ (x) tan² (x) - cos⁴ (x) tan⁴ (x)

0 = cos⁴ (x) [1] + cos⁴ (x) [2tan² (x) ] + cos⁴ (x) [tan⁴ (x) ]

0 = cos⁴ (x) [1 + 2tan² (x) + tan⁴ (x) ]

0 = cos⁴ (x) [1 + tan² (x) + tan² (x) + tan⁴ (4) ]

0 = cos⁴ (x) [1 (1) + 1 (tan² (x)) + tan² (x) (1) + tan² (x) (tan² (x) ]

0 = cos⁴ (x) [1 (1 + tan² (x)) + tan² (x) (1 + tan² (x)) ]

0 = cos⁴ (x) (1 + tan² (x)) (1 + tan² (x))

0 = cos⁴ (x) (1 + tan² (x)) ²

0 = cos⁴ (x) or 0 = (1 + tan² (x)) ²

⁴√0 = ⁴√cos⁴ (x) or √0 = (√1 + tan² (x)) ²

0 = cos (x) or 0 = 1 + tan² (x)

cos⁻¹ (0) = cos⁻¹ (cos (x)) or - 1 = tan² (x)

90 = x or √-1 = √tan² (x)

i = tan (x)

(No Solution)

2. sin (x) [tan (x) cos (x) - cot (x) cos (x) ] = 1 - 2cos² (x)

sin (x) [sin (x) - cos (x) cot (x) ] = 1 - cos² (x) - cos² (x)

sin (x) [sin (x) ] - sin (x) [cos (x) cot (x) ] = sin² (x) - cos² (x)

sin² (x) - cos² (x) = sin² (x) - cos² (x)

+ cos² (x) + cos² (x)

sin² (x) = sin² (x)

- sin² (x) - sin² (x)

0 = 0

3. 1 + sec² (x) sin² (x) = sec² (x)

sec² (x) sec² (x)

cos² (x) + sin² (x) = 1

cos² (x) = 1 - sin² (x)

√cos² (x) = √ (1 - sin² (x))

cos (x) = √ (1 - sin² (x))

cos⁻¹ (cos (x)) = cos⁻¹ (√1 - sin² (x))

x = 0

4. - tan² (x) + sec² (x) = 1

-1 - 1

tan² (x) - sec² (x) = - 1

tan² (x) = - 1 + sec²

√tan² (x) = √ (-1 + sec² (x))

tan (x) = √ (-1 + sec² (x))

tan⁻¹ (tan (x)) = tan⁻¹ (√ (-1 + sec² (x))

x = 0