Archive
6th of April 2017
Mathematics
Q:

tan a + sin a = m & tan a - sin a = n prove that m2 - n2 4square root of m n end root

Vinay Rockers - ICSE - Class X

Thursday, April 06, 2017 at 13:47:PM

A:

Given that tan a + sin a = m and tan a - sin a = n 

Consider,

mn = (tan a + sin a) (tan a - sin a) 

mn = tan2 a - sin2 a =begin mathsize 12px style fraction numerator sin squared straight a over denominator space cos squared space straight a end fraction minus sin squared straight a end style

mn = begin mathsize 12px style sin squared space straight a open parentheses fraction numerator 1 over denominator space cos squared space straight a end fraction minus 1 close parentheses end style

mn = begin mathsize 12px style sin squared space straight a open parentheses fraction numerator 1 minus space cos squared space straight a over denominator space cos squared space straight a end fraction close parentheses end style  

mn = tan2 a  × sin2 a      .......( i ) 

Consider,

LHS = m2 - n2

        = (tan a + sin a )2 - (tan a - sin a)2 

        = tan2 a  + sin2 a   + 2 tan A sin A - tan2 a   - sin2 a  + 2 tan A sin A 

        = 4 tan A sin A 

        = 4square root of mn [from equation ( i )] 
        = RHS

Hence proved.

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Thursday, April 06, 2017 at 14:11:PM