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Inverse Trigonometric Functions

Short Answer Type

51. Prove that

cos to the power of negative 1 end exponent x minus cos to the power of negative 1 end exponent y equals cos to the power of negative 1 end exponent open square brackets x space y plus square root of left parenthesis 1 minus x squared right parenthesis left parenthesis 1 minus y squared right parenthesis end root close square brackets

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52. Prove that : $tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses equals straight pi over 4$

straight L. straight H. straight S. equals space tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 fifth close parentheses plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals open square brackets tan to the power of negative 1 end exponent open parentheses 1 half close parentheses plus tan to the power of negative 1 end exponent open parentheses 15 over blank close parentheses close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style 1 half end style plus begin display style 1 fifth end style over denominator 1 minus open parentheses begin display style 1 half end style close parentheses open parentheses begin display style 1 fifth end style close parentheses end fraction close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator 1 minus xy end fraction close parentheses close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator 5 plus 2 over denominator 10 minus 1 end fraction close square brackets plus tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses equals tan to the power of negative 1 end exponent open parentheses 1 over 8 close parentheses space space space space space space space space space space equals space tan to the power of negative 1 end exponent space open square brackets fraction numerator begin display style 7 over 9 end style plus begin display style 1 over 8 end style over denominator 1 minus open parentheses begin display style 7 over 9 end style close parentheses open parentheses begin display style 1 over 8 end style close parentheses end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 56 plus 9 over denominator 72 minus 7 end fraction close square brackets equals tan to the power of negative 1 end exponent open parentheses 65 over 65 close parentheses space space space space space space space space space space equals space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis equals straight pi over 4 equals straight R. straight H. straight S.

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53.

Prove space that space tan to the power of negative 1 end exponent 1 fifth plus tan to the power of negative 1 end exponent 1 over 7 plus tan to the power of negative 1 end exponent 1 third plus tan to the power of negative 1 end exponent 1 over 8 equals straight pi over 4.

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54.

Prove that $tan to the power of negative 1 end exponent space straight t plus tan to the power of negative 1 end exponent fraction numerator 2 space straight l over denominator 1 minus straight t squared end fraction equals tan to the power of negative 1 end exponent fraction numerator 3 straight t minus straight t cubed over denominator 1 minus 3 straight t squared end fraction. straight t greater than 0$

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55.

Prove that $tan to the power of negative 1 end exponent 1 fourth plus tan to the power of negative 1 end exponent 2 over 9 equals 1 half cos to the power of negative 1 end exponent 3 over 5$

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56.

Prove that $tan to the power of negative 1 end exponent 1 over 7 plus tan to the power of negative 1 end exponent 2 over 9 equals 1 half cos to the power of negative 1 end exponent 3 over 5$

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Long Answer Type

57.

Prove that $cot to the power of negative 1 end exponent open parentheses fraction numerator ab plus 1 over denominator straight a minus straight b end fraction close parentheses plus cot to the power of negative 1 end exponent open parentheses fraction numerator bc plus 1 over denominator straight b minus straight c end fraction close parentheses plus cot to the power of negative 1 end exponent open parentheses fraction numerator ca plus 1 over denominator straight c minus straight a end fraction close parentheses equals 0$