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371. Prove the following identity:

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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372. Prove the following identity:

tan squared straight A minus tan squared straight B space equals space fraction numerator sin squared straight A minus sin squared straight B over denominator cos squared straight A cross times cos squared straight B end fraction.

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373. Prove the following identity:

left parenthesis secA minus tanA right parenthesis squared space left parenthesis 1 plus sinA right parenthesis space equals space 1 minus space sinA.

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374. Prove the following identity:

sec to the power of 6 straight theta space equals space tan to the power of 6 straight theta space plus space 3 space tan space straight theta. space secθ plus 1

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375. Prove the following identity:

left parenthesis 1 plus tan squared straight theta right parenthesis plus open parentheses 1 plus fraction numerator 1 over denominator tan squared straight theta end fraction close parentheses space equals fraction numerator 1 over denominator sin squared straight theta minus sin to the power of 4 straight theta end fraction

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376. Prove the following identity:

left parenthesis 1 plus tanA. tanB right parenthesis squared plus left parenthesis tanA minus tanB right parenthesis squared space equals sec squared straight A. space sec squared straight B.

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377. Prove the following identity:

cotθ minus tanθ space equals space fraction numerator 2 space cos squared straight theta minus 1 over denominator sinθ minus cosθ end fraction.

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378. Prove the following identity:

tan squared straight theta plus cot squared straight theta plus 2 space equals space sec squared straight theta. space cosec squared straight theta.

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379. Prove the following identity:

left parenthesis 1 plus cotA plus tanA right parenthesis space left parenthesis sinA minus cosA right parenthesis space equals space sinA. space tanA minus space cotA. space cosA.

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380. If 7 sin²θ + 3cos²θ = 4, show that

tanθ space equals space fraction numerator 1 over denominator square root of 3 end fraction.

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