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Introduction to Trigonometry

Short Answer Type

51. Prove the following identities where the angles involved are acute angles for which the expressions are defined.
(sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A

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52. Prove the following identities where the angles involved are acute angles for which the expressions are defined.
(cosec A - sin A) (sec A - cos A) =

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Long Answer Type

53. Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$open parentheses fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction close parentheses space equals space open parentheses fraction numerator 1 minus tanA over denominator 1 minus cotA end fraction close parentheses squared$

open parentheses fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction close parentheses space equals space open parentheses fraction numerator 1 minus tanA over denominator 1 minus cotA end fraction close parentheses squared

L.H.S. =

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

space space equals space fraction numerator sec squared straight A over denominator cosec squared straight A end fraction space equals space fraction numerator begin display style fraction numerator 1 over denominator cos squared straight A end fraction end style over denominator begin display style fraction numerator 1 over denominator sin squared straight A end fraction end style end fraction space space space space equals space fraction numerator 1 over denominator cos squared straight A end fraction cross times fraction numerator sin squared straight A over denominator 1 end fraction space space space equals space fraction numerator sin squared straight A over denominator cos squared straight A end fraction space equals space tan squared straight A

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Short Answer Type

54. Write the value of sin² 69° + sin² 21°.

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55. If sin θ = 1/3, then find the value of (2 cot²θ+ 2).

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56. In ΔABC, ∠B = 90°
3 sin A = 3/5 , what is the value cos C.

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57. If 15 cot θ = 8, then find the value of

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58.

Simplify: $fraction numerator sin cubed straight theta plus cos cubed straight theta over denominator sinθ plus cosθ end fraction plus sinθ. space cosθ$

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59. Write the value of sin (65° + θ) -cos (25° - θ).

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60. Write the value of

fraction numerator 5 over denominator cot squared straight theta end fraction minus fraction numerator 5 over denominator cos squared straight theta end fraction.

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