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222. Prove the following identities:

fraction numerator sin space straight theta over denominator cot space straight theta plus cosecθ end fraction space equals space 2 plus fraction numerator sinθ over denominator cotθ minus cosecθ end fraction.

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223. Prove the following identities:
sin⁶ θ + cos⁶ θ + 3 sin² θ . cos² θ = 1

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224. Prove the following identities:
(tan A + cosec B)² - (cot B - sec A)² = 2 tan A.cot B (cosec A + sec B)

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225. Prove the following identities:
(sin⁸ θ - cos⁸ θ) = (sin² θ - cos² θ) (1 - 2 sin² θ . cos² θ)

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226. Prove the following identities:

cot squared straight A open parentheses fraction numerator secA minus 1 over denominator 1 plus sinA end fraction close parentheses plus sec squared straight A open parentheses fraction numerator sinA minus 1 over denominator 1 plus secA end fraction close parentheses equals 0

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227. Prove the following identities:

fraction numerator cosA over denominator 1 minus sinA end fraction plus fraction numerator sinA over denominator 1 minus cosA end fraction plus 1 space equals space fraction numerator sinA. space cosA over denominator left parenthesis 1 minus sinA right parenthesis space left parenthesis 1 minus cosA right parenthesis end fraction.

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228. Prove the following identities:
(sin⁴ θ + cos⁴ θ) = 1 - 2 sin² θ . cos² θ

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229. Prove the following identities:
(sin⁴ θ - sec² θ) = tan⁴ θ + tan² θ

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230. Prove the following identities:
(sin⁴ θ - cos⁴ θ) = sin² θ - cos² θ = 2 sin² θ-1 = 1- 2 cos² θ.

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