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Introduction to Trigonometry

Short Answer Type

41. Choose the correct option and Justify your choice.

left parenthesis 1 plus tanθ plus secθ right parenthesis space left parenthesis 1 plus cotθ minus cosecθ right parenthesis

=
(A) 0 (B) 1 (C) 2 (D) -1

128 Views

42. Choose the correct option and Justify your choice.
(sec A + tan A) (1 - sin A) =
(A) sec A (B) sin A (C) cosec A (D) cos A

104 Views

43. Choose the correct option and Justify your choice.

space fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction equals

(A)

sec squared straight A

(B) -1 (C)

cot squared straight A

(D)

tan squared straight A

89 Views

44. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

space space left parenthesis cosecθ space minus space cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction.

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45. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

fraction numerator cosA over denominator 1 plus sinA end fraction plus fraction numerator 1 plus sinA over denominator cosA end fraction equals space 2 space secA.

99 Views

Long Answer Type

46. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cotθ over denominator 1 minus tanθ end fraction space equals space 1 plus secθ. space cosecθ

147 Views

Short Answer Type

47. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

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121 Views

Long Answer Type

48. Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$fraction numerator cos space straight A minus sin space straight A plus 1 over denominator cos space straight A plus sin space straight A minus 1 end fraction equals cosec space straight A plus cot space straight A.$

fraction numerator cosA minus sinA plus 1 over denominator cosA plus sinA minus 1 end fraction space equals space cosecA plus cotA

L.H.S. =

fraction numerator cosA minus sinA plus 1 over denominator cosA plus sinA minus 1 end fraction

Dividing N' and D' by sin A, we get

space equals space fraction numerator begin display style fraction numerator cosA minus sinA plus 1 over denominator sinA end fraction end style over denominator begin display style fraction numerator cosA plus sinA minus 1 over denominator sinA end fraction end style end fraction

space space space space equals space fraction numerator begin display style cosA over sinA minus sinA over sinA plus 1 over sinA end style over denominator begin display style cosA over sinA plus sinA over sinA minus 1 over sinA end style end fraction

equals space fraction numerator cotA minus 1 plus cosecA over denominator cotA plus 1 minus cosecA end fraction equals space fraction numerator cotA plus cosecA minus 1 over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosec squared straight A minus cot squared straight A right parenthesis over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosecA plus cotA right parenthesis space left parenthesis cosecA minus cotA right parenthesis over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis left parenthesis 1 minus cosecA plus cotA right parenthesis over denominator left parenthesis cotA minus cosecA plus 1 right parenthesis end fraction equals space cot space straight A space plus space cosec space straight A space equals space straight R. straight H. straight S. space Hence comma space straight L. straight H. straight S. space equals space straight R. straight H. straight S.

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49. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

square root of fraction numerator 1 plus sinA over denominator 1 minus sinA end fraction end root space equals space secA plus tanA.

118 Views

Short Answer Type

50. Prove the following identities where the angles involved are acute angles for which the expressions are defined.

fraction numerator sinθ minus 2 sin cubed straight theta over denominator 2 cos cubed straight theta minus cosθ end fraction equals space space tanθ

103 Views