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

L.H.S
$equals space left parenthesis 1 plus tanA. space tanB right parenthesis squared plus left parenthesis tanA minus tanB right parenthesis squared equals space left parenthesis 1 plus tan squared straight A. space tan squared straight B space plus space 2 space tanA. space tanB space plus space tan squared straight A space plus space tan squared straight B space minus space 2 space tanA. space tanB right parenthesis equals space 1 plus tan squared straight A plus tan squared straight B plus tan squared straight A. space tan squared straight B equals space left parenthesis 1 plus tan squared straight A right parenthesis plus space tan squared straight B left parenthesis 1 plus tan squared straight A right parenthesis equals space left parenthesis 1 plus tan squared straight B right parenthesis thin space left parenthesis 1 plus tan squared straight A right parenthesis equals space sec squared straight B. space sec squared straight A equals space straight R. straight H. straight S.$
Hence, L.H.S. = R.H.S.