Prove the following identity: from Mathematics Introduction to

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Introduction to Trigonometry

Short Answer Type

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

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

L.H.S. =

cot space straight theta space minus space tanθ

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

Hence, L.H.S. = R.H.S.

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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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Long Answer Type

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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Short Answer Type

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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