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211. Prove the following identities:

fraction numerator cosθ over denominator cosecθ plus 1 end fraction plus fraction numerator cosθ over denominator cosecθ minus 1 end fraction equals space space 2 tanθ

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212. Prove the following identities:

fraction numerator left parenthesis 1 minus sinθ right parenthesis squared plus left parenthesis 1 minus sinθ right parenthesis squared over denominator 2 space cos squared straight theta end fraction space equals space fraction numerator 1 plus sin squared straight theta over denominator 1 minus sin squared straight theta end fraction.

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213. Prove the following identities:

square root of fraction numerator 1 plus sinθ over denominator 1 minus sinθ end fraction end root space equals space secθ plus tanθ

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214. Prove the following identities:

square root of fraction numerator 1 plus cosβ over denominator 1 minus cosβ end fraction end root space equals space cosecβ plus cotβ

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215. Prove the following identities:

square root of fraction numerator 1 minus sinθ over denominator 1 plus sinθ end fraction end root equals space space secθ plus tanθ

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216. Prove the following identities:

square root of fraction numerator secθ minus 1 over denominator secθ plus 1 end fraction end root plus square root of fraction numerator secθ plus 1 over denominator secθ minus 1 end fraction end root space equals space 2 space cosecθ

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217. Prove the following identities:

square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root plus square root of fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction end root space equals space 2 space cosec space straight theta

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Solution not provided.

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219. Prove the following identities:

fraction numerator sinθ plus cosθ over denominator sinθ minus cosθ end fraction plus fraction numerator sinθ minus cosθ over denominator sinθ plus cosθ end fraction space equals space fraction numerator 2 over denominator sin squared straight theta plus cos squared straight theta end fraction equals fraction numerator 2 over denominator 2 sin squared straight theta minus 1 end fraction

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220. Prove the following identities:

fraction numerator sinA minus sinB over denominator cosA plus cosB end fraction plus fraction numerator cosA minus cosB over denominator sinA plus sinB end fraction equals 0

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