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Inverse Trigonometric Functions

Short Answer Type

41.

Prove the following :

$3 space sin to the power of negative 1 end exponent straight x equals sin to the power of negative 1 end exponent left parenthesis 3 straight x minus 4 straight x cubed right parenthesis comma space straight x element of open square brackets negative 1 half comma 1 half close square brackets$

148 Views

42.

Prove the following :

$3 space cos to the power of negative 1 end exponent straight x equals cos to the power of negative 1 end exponent thin space left parenthesis 4 straight x cubed minus 3 straight x right parenthesis comma space straight x element of open square brackets 1 half comma space 1 close square brackets$

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43. $Prove space that space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 space sin to the power of negative 1 end exponent 1 third equals 9 over 4 sin to the power of negative 1 end exponent fraction numerator 2 square root of 2 over denominator 3 end fraction.$

straight L. straight H. straight S. space equals space fraction numerator 9 straight pi over denominator 8 end fraction minus 9 over 4 sin to the power of negative 1 end exponent open parentheses 1 third close parentheses equals 9 over 4 open curly brackets straight pi over 2 minus sin to the power of negative 1 end exponent 1 third close curly brackets space space space space space space space space space space equals space 9 over 4 space open curly brackets cos to the power of negative 1 end exponent open parentheses 1 third close parentheses close curly brackets 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 sin to the power of negative 1 end exponent space straight x plus space cos to the power of negative 1 end exponent space straight x equals straight x over 2 close square brackets space space space space space space space space space space equals 9 over 4 space sin to the power of negative 1 end exponent square root of 1 minus open parentheses 1 third close parentheses squared end root space space space space space space space space space space space space left square bracket because space cos to the power of negative 1 end exponent space straight x space equals space sin to the power of negative 1 end exponent space square root of 1 minus straight x squared end root space for space 0 less or equal than straight x space less or equal than space 1 space right square bracket space space space space space space space space space space equals space 9 over 4 space sin to the power of negative 1 end exponent space open parentheses square root of 8 over 9 end root close parentheses equals 9 over 4 space sin to the power of negative 1 end exponent space open parentheses fraction numerator 2 square root of 2 over denominator 3 end fraction close parentheses equals space straight R. straight H. straight S.

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44.

Show that $tan to the power of negative 1 end exponent 1 half plus tan to the power of negative 1 end exponent 2 over 11 equals tan to the power of negative 1 end exponent 3 over 4.$

107 Views

45.

Prove that $sin to the power of negative 1 end exponent space straight x plus space cos to the power of negative 1 end exponent space straight x equals straight pi over 2$

170 Views

46.

Prove that $tan to the power of negative 1 end exponent straight x plus cot to the power of negative 1 end exponent straight x equals straight pi over 2 comma space straight x element of straight R$

116 Views

47.

Prove that $cos e c to the power of negative 1 end exponent x plus s e c to the power of negative 1 end exponent x equals straight pi over 2 comma space open vertical bar x close vertical bar greater or equal than 1$