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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

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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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We have,

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Now,

space fraction numerator left parenthesis 1 minus sinθ right parenthesis thin space left parenthesis 1 plus sin space straight theta right parenthesis over denominator left parenthesis 1 plus cosθ right parenthesis thin space left parenthesis 1 minus cosθ right parenthesis end fraction space equals space fraction numerator 1 minus sin squared straight theta over denominator 1 minus cos squared straight theta end fraction

equals space fraction numerator cos squared straight theta over denominator sin squared straight theta end fraction space equals space cot squared straight theta space equals space open parentheses 8 over 15 close parentheses squared space equals space 64 over 225 Ans.

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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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