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61. Solve :

tan to the power of negative 1 end exponent open parentheses fraction numerator 1 plus x over denominator 1 minus x end fraction close parentheses equals straight pi over 4 plus tan to the power of negative 1 end exponent comma space x comma space o less than x less than 1.

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62. Solve, the following equations;
2 tan^–1(cos x) = tan–^1t (2 cosec x)

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63. Solve, the following equations;

tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus x over denominator 1 plus x end fraction close parentheses equals 1 half tan to the power of negative 1 end exponent space x comma space left parenthesis x space greater than space 0 right parenthesis

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64.

cos to the power of negative 1 end exponent x plus sin to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 5 end fraction equals straight pi over 4

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65.

tan to the power of negative 1 end exponent fraction numerator x minus 1 over denominator x minus 2 end fraction plus tan to the power of negative 1 end exponent fraction numerator x plus 1 over denominator x plus 2 end fraction equals straight pi over 4.

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66. Find the value of

tan open parentheses space sin to the power of negative 1 end exponent 3 over 5 plus cot to the power of negative 1 end exponent 3 over 2 close parentheses

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67. Prove that

2 space sin to the power of negative 1 end exponent 3 over 5 equals tan to the power of negative 1 end exponent 24 over 7.

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Let space sin to the power of negative 1 end exponent 3 over 5 equals straight x space and space sin to the power of negative 1 end exponent 8 over 17 equals straight y therefore space space space space space space sin space straight x equals 3 over 5 space and space sin space straight y space equals space 8 over 17 Now space space space space cos space straight x equals square root of 1 minus sin squared space straight x end root equals square root of 1 minus 9 over 25 end root equals square root of 16 over 25 end root equals 4 over 5 and space space space space space space cos space straight y equals square root of 1 minus sin squared space straight y end root equals square root of 1 minus 64 over 289 end root equals square root of 225 over 289 end root equals 15 over 17

Now cos(x - y) = cos x cos y + sin x sin y

equals 4 over 5 cross times 15 over 17 plus 3 over 5 cross times 8 over 17 equals 60 over 85 plus 24 over 85 equals fraction numerator 60 plus 24 over denominator 85 end fraction equals 84 over 85 therefore space space space space space straight x minus straight y equals cos to the power of negative 1 end exponent 84 over 85 sin to the power of negative 1 end exponent 3 over 5 asterisk times sin to the power of negative 1 end exponent 8 over 17 minus cos to the power of negative 1 end exponent 84 over 85

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69. Prove that

cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 33 over 65.

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70. Prove that

cos to the power of negative 1 end exponent 12 over 13 plus sin to the power of negative 1 end exponent 3 over 5 equals sin to the power of negative 1 end exponent 56 over 65.

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