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Introduction to Trigonometry

Short Answer Type

381. Prove that: sinθ (1 + tanθ) + cosθ (1 + cot θ) = secθ + cosecθ

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382.

Prove that:
$sec squared straight theta minus fraction numerator sin squared straight theta minus 2 sin to the power of 4 straight theta over denominator 2 space cos to the power of 4 straight theta space minus cos squared straight theta end fraction equals 1$

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Long Answer Type

383.

If $fraction numerator tan space straight A over denominator tan space straight B end fraction equals straight n space and space fraction numerator sin space straight A over denominator sin space straight B end fraction equals straight m comma$ then show that $cos squared straight A space equals space fraction numerator straight m squared minus 1 over denominator straight n squared minus 1 end fraction.$

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384.

Prove the following identity:
$fraction numerator 1 over denominator secθ minus tanθ end fraction minus 1 over cosθ space equals space 1 over cosθ minus fraction numerator 1 over denominator secθ plus tanθ end fraction.$

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385.
Prove the following identity:
$left parenthesis secθ minus cosecθ right parenthesis space left parenthesis 1 plus tanθ plus cotθ right parenthesis space equals space secθ. space tanθ space minus space cosecθ. space cotθ$

straight L. straight H. straight S. space equals space left parenthesis secθ minus cosecθ right parenthesis space left parenthesis 1 plus tanθ plus space cotθ right parenthesis equals space open parentheses 1 over cosθ minus 1 over sinθ close parentheses space open parentheses 1 plus sinθ over cosθ plus cosθ over sinθ close parentheses equals space open parentheses fraction numerator sinθ minus cosθ over denominator cosθ. space sinθ end fraction close parentheses space open parentheses fraction numerator sinθ. space cosθ space plus sin squared straight theta plus cos squared straight theta over denominator cosθ. space sinθ end fraction close parentheses equals space fraction numerator sin cubed straight theta minus cos cubed straight theta over denominator sin squared straight theta. space cos squared straight theta end fraction equals space fraction numerator sin cubed straight theta over denominator sin squared straight theta. space cos squared straight theta end fraction minus fraction numerator cos cubed straight theta over denominator sin squared straight theta. space cos squared straight theta end fraction equals space fraction numerator sin space straight theta over denominator cos squared straight theta end fraction minus fraction numerator cosθ over denominator sin squared straight theta end fraction equals space fraction numerator sin space straight theta over denominator cos space straight theta end fraction cross times fraction numerator 1 over denominator cos space straight theta end fraction minus fraction numerator cos space straight theta over denominator sin space straight theta end fraction cross times fraction numerator 1 over denominator sin space straight theta end fraction equals space tanθ. space secθ space minus space cot space straight theta. space cosec space straight theta equals space straight R. straight H. straight S.

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

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Short Answer Type

386.

Prove the following identity:
$fraction numerator tanθ plus secθ minus 1 over denominator tanθ minus secθ plus 1 end fraction equals fraction numerator 1 plus sinθ over denominator cosθ end fraction$

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387.

Prove the following identity:
$fraction numerator cosA over denominator 1 minus tanA end fraction minus fraction numerator sin squared straight A over denominator cosA minus sinA end fraction equals sinA plus cosA.$

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388.

Prove the following identity:
sec A (1 - sin A) (sec A + tan A) = 1

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389.

Prove the following identity:
$fraction numerator 1 over denominator secθ minus 1 end fraction plus fraction numerator 1 over denominator secθ plus 1 end fraction equals 2 space cosecθ. space cotθ.$

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390.

Prove the following identity:
$fraction numerator cosecθ plus cotθ over denominator cosecθ minus cotθ end fraction space equals space 1 plus 2 cot squared straight theta plus 2 cosecθ. space cotθ$