Question

Please solve this and only answer with image

Answer 1

`\cot^4 A - \cot^2 A = 1\\\\\cot^4 A = 1 + \cot^2 A\\\\(\cos^4 A)/(\sin^4 A) = 1 + (\cos^2 A)/(\sin^2 A)\\\\(\cos^4 A)/(\sin^4 A) = (\sin^2 A)/(\sin^2 A)+(\cos^2 A)/(\sin^2 A)\\\\(\cos^4 A)/(\sin^4 A) = (\sin^2 A+\cos^2 A)/(\sin^2 A)\\\\(\cos^4 A)/(\sin^4 A) = (1)/(\sin^2 A)\\\\\cos^4 A = (\sin^4 A)/(\sin^2 A)\\\\\cos^4 A = \sin^2 A\\\\`

This then means,

`\cos^4 A + \cos^2 A = 1\\\\\sin^2 A + \cos^2 A = 1\\\\`

which is the pythagorean trig identity. This concludes the proof.

Therefore, if
`\cot^4 A - \cot^2 A = 1`, then
`\cos^4 A + \cos^2 A = 1`