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Inverse Trigonometric Functions

Short Answer Type

71. Show that

sin to the power of negative 1 end exponent 12 over 13 plus cos to the power of negative 1 end exponent 4 over 5 plus tan to the power of negative 1 end exponent 63 over 16 equals straight pi.

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72. Prove that :

tan to the power of negative 1 end exponent 63 over 16 equals sin to the power of negative 1 end exponent 5 over 13 plus cos to the power of negative 1 end exponent 3 over 5

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73. Simplify $tan to the power of negative 1 end exponent open square brackets fraction numerator straight a space cos space straight x minus straight b space sin space straight x over denominator straight b space cos space straight x plus straight a space sin space straight x end fraction close square brackets comma space if space straight a over straight b space tan space straight x greater than negative 1.$

tan to the power of negative 1 end exponent open square brackets fraction numerator straight a space cos space straight x minus straight b space sin space straight x over denominator straight b space cos space straight x plus straight a space sin space straight x end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator straight a space cos space straight x minus straight b space sin space straight x over denominator straight b space cos space straight x end fraction end style over denominator begin display style fraction numerator straight b space cos space straight x plus straight a space sin space straight x over denominator straight b space cos space straight x end fraction end style end fraction close square brackets equals space tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style straight a over straight b end style minus tan space straight x over denominator 1 plus begin display style straight a over straight b end style space tan space straight x end fraction close square brackets equals tan to the power of negative 1 end exponent straight a over straight b minus tan to the power of negative 1 end exponent left parenthesis tan space straight x right parenthesis equals tan to the power of negative 1 end exponent straight a over straight b minus straight x.

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74.

Shoe that $sin to the power of negative 1 end exponent open parentheses 2 space x square root of 1 minus x squared end root close parentheses equals 2 space sin to the power of negative 1 end exponent x comma space space minus fraction numerator 1 over denominator square root of 2 end fraction less or equal than space x space less or equal than fraction numerator 1 over denominator square root of 2 end fraction$

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75.

Shoe that $sin to the power of negative 1 end exponent open parentheses 2 space x square root of 1 minus x squared end root close parentheses equals 2 space cos to the power of negative 1 end exponent space x comma space fraction numerator 1 over denominator square root of 2 end fraction space less or equal than x less or equal than 1$

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76.

Prove $sin space left parenthesis 2 space sin to the power of negative 1 end exponent space straight x right parenthesis equals space 2 space straight x square root of 1 minus straight x squared end root$

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77.

Prove $cos to the power of negative 1 end exponent space x equals 2 space sin to the power of negative 1 end exponent space square root of fraction numerator 1 minus x over denominator 2 end fraction end root equals 2 space cos to the power of negative 1 end exponent square root of fraction numerator 1 plus x over denominator 2 end fraction end root$