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Asked by bhavya13schs 12th April 2019, 9:25 AM
Answered by Expert
Answer:
C o n s i d e r comma
I equals fraction numerator cos open parentheses 2 pπ plus-or-minus begin display style straight pi over 4 end style close parentheses over denominator sin open curly brackets q straight pi plus left parenthesis negative 1 right parenthesis to the power of straight q times begin display style straight pi over 4 end style close curly brackets end fraction
equals fraction numerator cos left parenthesis 2 p straight pi right parenthesis space cos begin display style straight pi over 4 end style space minus-or-plus sin left parenthesis 2 pπ right parenthesis space sin straight pi over 4 over denominator sin left parenthesis q straight pi right parenthesis space cos open parentheses left parenthesis negative 1 right parenthesis to the power of straight q times straight pi over 4 close parentheses plus space cos left parenthesis qπ right parenthesis space sin open parentheses left parenthesis negative 1 right parenthesis to the power of straight q times straight pi over 4 close parentheses end fraction
equals fraction numerator cos left parenthesis 2 p straight pi right parenthesis space cos straight pi over 4 over denominator cos left parenthesis qπ right parenthesis space sin open parentheses left parenthesis negative 1 right parenthesis to the power of straight q times straight pi over 4 close parentheses end fraction space........ space S i n c e space sin left parenthesis n straight pi right parenthesis equals 0 space f o r space e v e r y space n element of straight integer numbers
equals fraction numerator cos straight pi over 4 over denominator cos left parenthesis qπ right parenthesis space sin open parentheses left parenthesis negative 1 right parenthesis to the power of straight q times straight pi over 4 close parentheses end fraction space........ space S i n c e space cos left parenthesis 2 n straight pi right parenthesis equals 1 space f o r space e v e r y space n element of straight integer numbers
equals fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis space sin straight pi over 4 end fraction space......... space S i n c e space sin left parenthesis negative x right parenthesis equals negative sin space x
equals fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis space fraction numerator 1 over denominator square root of 2 end fraction end fraction space
equals fraction numerator begin display style 1 end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis end fraction
rightwards double arrow I equals fraction numerator begin display style 1 end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis end fraction
N o w comma space i f space q space i s space e v e n space t h e n space w e space h a v e
I equals fraction numerator begin display style 1 end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis end fraction equals fraction numerator 1 over denominator left parenthesis 1 right parenthesis space cos left parenthesis 2 n straight pi right parenthesis end fraction equals 1 space...... space S i n c e space cos left parenthesis 2 n straight pi right parenthesis equals 1 space f o r space e v e r y space n element of straight integer numbers
A n d space i f space q space i s space o d d comma space t h e n space w e space h a v e
I equals fraction numerator begin display style 1 end style over denominator left parenthesis negative 1 right parenthesis to the power of q space cos left parenthesis qπ right parenthesis end fraction equals fraction numerator 1 over denominator left parenthesis negative 1 right parenthesis space cos left parenthesis 2 n plus 1 right parenthesis straight pi end fraction equals fraction numerator 1 over denominator left parenthesis negative 1 right parenthesis left parenthesis negative 1 right parenthesis end fraction equals 1 space...... space space S i n c e space cos left parenthesis 2 n plus 1 right parenthesis straight pi equals negative 1 space f o r space e v e r y space n element of straight integer numbers
S o comma space i n space b o t h space t h e space c a s e s space w e space a r e space g e t t i n g space I equals 1
H e n c e comma space fraction numerator cos open parentheses 2 pπ plus-or-minus begin display style straight pi over 4 end style close parentheses over denominator sin open curly brackets q straight pi plus left parenthesis negative 1 right parenthesis to the power of straight q times begin display style straight pi over 4 end style close curly brackets end fraction equals 1.
Answered by Expert 12th April 2019, 9:58 AM
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