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Asked by eknathmote50 | 02 Jun, 2019, 14:19: PM
answered-by-expert Expert Answer
f left parenthesis x right parenthesis equals tan open parentheses sin to the power of negative 1 end exponent 2 x close parentheses
W e space k n o w space t h a t space sin to the power of negative 1 end exponent x element of open square brackets negative straight pi over 2 comma straight pi over 2 close square brackets space a n d space x element of open square brackets negative 1 comma 1 close square brackets
rightwards double arrow sin to the power of negative 1 end exponent 2 x element of open square brackets negative straight pi over 2 comma straight pi over 2 close square brackets space a n d space 2 x element of open square brackets negative 1 comma 1 close square brackets
rightwards double arrow tan open parentheses sin to the power of negative 1 end exponent 2 x close parentheses element of open square brackets tan open parentheses negative straight pi over 2 close parentheses comma tan open parentheses straight pi over 2 close parentheses close square brackets space a n d space x element of open square brackets negative 1 half comma 1 half close square brackets
rightwards double arrow tan open parentheses sin to the power of negative 1 end exponent 2 x close parentheses element of open parentheses negative infinity comma space infinity close parentheses space a n d space x element of open square brackets negative 1 half comma 1 half close square brackets
S o space d o m a i n space o f space f left parenthesis x right parenthesis space i s space open square brackets negative 1 half comma 1 half close square brackets space a n d space r a n g e space o f space f left parenthesis x right parenthesis space i s space open parentheses negative infinity comma space infinity close parentheses
N o w comma space l e t space sin to the power of negative 1 end exponent 2 x equals theta rightwards double arrow 2 x equals sin theta
A l s o comma space
tan theta equals fraction numerator sin theta over denominator cos theta end fraction equals fraction numerator sin theta over denominator square root of 1 minus sin squared theta end root end fraction equals fraction numerator 2 x over denominator square root of 1 minus open parentheses 2 x close parentheses squared end root end fraction equals fraction numerator 2 x over denominator square root of 1 minus 4 x squared end root end fraction
S i n c e comma theta equals space sin to the power of negative 1 end exponent 2 x
rightwards double arrow tan open parentheses sin to the power of negative 1 end exponent 2 x close parentheses equals fraction numerator 2 x over denominator square root of 1 minus 4 x squared end root end fraction
H e n c e comma space f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator square root of 1 minus 4 x squared end root end fraction
f apostrophe left parenthesis x right parenthesis equals s e c squared open parentheses sin to the power of negative 1 end exponent 2 x close parentheses times fraction numerator 1 over denominator square root of 1 minus 4 x squared end root end fraction
f apostrophe open parentheses 1 fourth close parentheses equals s e c squared open parentheses sin to the power of negative 1 end exponent open parentheses 2 times 1 fourth close parentheses close parentheses times fraction numerator 1 over denominator square root of 1 minus 4 open parentheses begin display style 1 fourth end style close parentheses squared end root end fraction
equals s e c squared open parentheses sin to the power of negative 1 end exponent open parentheses 1 half close parentheses close parentheses times fraction numerator 2 over denominator square root of 3 end fraction
equals s e c squared open parentheses straight pi over 6 close parentheses times fraction numerator 2 over denominator square root of 3 end fraction
equals open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared times fraction numerator 2 over denominator square root of 3 end fraction
equals fraction numerator square root of 3 over denominator 2 end fraction
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