Tan a=4/3; cos (a+b)= - √5/5 ; find the value of tan (a - b) .

Asked by saikatsukai514 | 17th Apr, 2020, 07:41: AM

Expert Answer:

Given: tan a = 4/3
rightwards double arrow c o t space a equals 3 over 4
sin space a equals fraction numerator 1 over denominator cos e c space a end fraction equals fraction numerator 1 over denominator square root of 1 plus c o t squared a end root end fraction equals fraction numerator 1 over denominator square root of 1 plus begin display style 9 over 16 end style end root end fraction equals 4 over 5
rightwards double arrow cos squared a equals 1 minus sin squared a equals 1 minus 16 over 25 equals 9 over 25
rightwards double arrow cos space a equals 3 over 5
cos left parenthesis a plus b right parenthesis equals negative fraction numerator square root of 5 over denominator 5 end fraction equals negative fraction numerator 1 over denominator square root of 5 end fraction
rightwards double arrow cos space a space cos space b minus sin space a space sin space b equals negative fraction numerator 1 over denominator square root of 5 end fraction
rightwards double arrow 3 over 5 cos space b minus 4 over 5 sin space b equals negative fraction numerator 1 over denominator square root of 5 end fraction
rightwards double arrow cos space b equals 5 over 3 open parentheses negative fraction numerator 1 over denominator square root of 5 end fraction plus 4 over 5 sin space b close parentheses space... space left parenthesis i right parenthesis
S i n c e comma space sin squared b equals 1 minus cos squared b
rightwards double arrow sin squared b equals 1 minus open square brackets 5 over 3 open parentheses negative fraction numerator 1 over denominator square root of 5 end fraction plus 4 over 5 sin space b close parentheses close square brackets squared
rightwards double arrow 25 over 9 sin squared b minus fraction numerator 40 over denominator 9 square root of 5 end fraction sin space b minus 4 over 9 equals 0
S o l v i n g space t h i s comma space w e space g e t
sin space b equals fraction numerator 2 over denominator square root of 5 end fraction space space o r space space minus fraction numerator 2 over denominator 5 square root of 5 end fraction
rightwards double arrow cos b equals square root of 1 minus si n squared b end root equals fraction numerator 1 over denominator square root of 5 end fraction space space o r space space fraction numerator 11 over denominator 5 square root of 5 end fraction
rightwards double arrow tan b equals 2 space space o r space space minus 2 over 11
rightwards double arrow tan left parenthesis a minus b right parenthesis equals fraction numerator tan space a minus tan space b over denominator 1 minus tan space a space tan space b end fraction equals fraction numerator begin display style 4 over 3 minus 2 end style over denominator 1 minus begin display style 4 over 3 end style cross times 2 end fraction equals fraction numerator 4 minus 6 over denominator 3 minus 8 end fraction equals 2 over 5 space space o r space space fraction numerator begin display style 4 over 3 plus 2 over 11 end style over denominator 1 minus begin display style 4 over 3 end style cross times begin display style 2 over 11 end style end fraction equals fraction numerator 44 plus 6 over denominator 33 minus 8 end fraction equals 2

Answered by Renu Varma | 17th Apr, 2020, 11:31: AM

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