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31. If tan 2A = cot (A - 18°), where 2A is an acute angle, find the value of A.

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32. If tan A = cot B, prove that A + B = 90°.

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33. If sec 4A = cosec (A - 20°), where 4A is an acute angle, find the value of A.

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34. If A, B and C are interior angles of ΔABC, then show that

space sin space open parentheses fraction numerator straight B plus straight C over denominator 2 end fraction close parentheses space equals space cos straight A over 2.

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35. Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

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36. Express the trigonometric ratios sin A sec A, and tan A in terms of cot A.

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37. Write all the other trigonometric ratios of ∠A in terms of sec A.

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38. Evaluate:

fraction numerator sin squared 63 degree plus sin squared 27 degree over denominator cos squared 17 degree plus cos squared 73 degree end fraction.

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sin (90° - 65°). cos 65° + cos (90° - 65°). sin 65°
= cos65°. cos 65° + sin 65° sin 65°
= cos² 65° + sin² 65° [Using sin² A + cos² A = 1]
= 1.

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40. Choose the correct option and Justify your choice.
9 sec²A - 9 tan²A =
(A) 1 (B) 9 (C) 8 (D) 0

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