If sin^4 theta/a + cos^4 theta/b = 1/ a+b; prove sin^8theta/ a^3 + cos^8 theta/ b^3 = 1/ (a+b)^3?
 
 

Asked by vans75a | 27th Aug, 2015, 10:08: AM

Expert Answer:

fraction numerator sin to the power of 4 theta over denominator a end fraction plus fraction numerator cos to the power of 4 theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow open parentheses a plus b close parentheses open square brackets fraction numerator s i n to the power of 4 theta over denominator a end fraction plus fraction numerator c o s to the power of 4 theta over denominator b end fraction close square brackets equals 1 rightwards double arrow open parentheses 1 plus b over a close parentheses s i n to the power of 4 theta plus open parentheses a over b plus 1 close parentheses c o s to the power of 4 theta equals 1 rightwards double arrow s i n to the power of 4 theta plus c o s to the power of 4 theta plus b over a s i n to the power of 4 theta plus a over b c o s to the power of 4 theta equals 1 rightwards double arrow b over a s i n to the power of 4 theta plus a over b c o s to the power of 4 theta equals open parentheses sin squared theta plus cos squared theta close parentheses squared minus open parentheses s i n to the power of 4 theta plus c o s to the power of 4 theta close parentheses rightwards double arrow b over a s i n to the power of 4 theta plus a over b c o s to the power of 4 theta equals 2 sin squared theta cos squared theta rightwards double arrow open parentheses square root of b over a end root sin squared theta close parentheses squared plus open parentheses square root of a over b end root cos squared theta close parentheses squared minus 2 s i n squared theta c o s squared theta equals 0 rightwards double arrow open parentheses square root of b over a end root s i n squared theta close parentheses squared plus open parentheses square root of a over b end root c o s squared theta close parentheses squared minus 2 square root of b over a end root s i n squared theta square root of a over b end root c o s squared theta equals 0 rightwards double arrow open parentheses square root of b over a end root s i n squared theta minus square root of a over b end root c o s squared theta close parentheses squared equals 0 rightwards double arrow square root of b over a end root s i n squared theta equals square root of a over b end root c o s squared theta rightwards double arrow fraction numerator s i n squared theta over denominator a end fraction equals fraction numerator c o s squared theta over denominator b end fraction rightwards double arrow fraction numerator s i n squared theta over denominator a end fraction equals fraction numerator c o s squared theta over denominator b end fraction equals fraction numerator s i n squared theta plus c o s squared theta over denominator a plus b end fraction equals fraction numerator 1 over denominator a plus b end fraction
 
H e n c e comma space L H S equals fraction numerator sin to the power of 8 theta over denominator a cubed end fraction plus fraction numerator cos to the power of 8 theta over denominator b cubed end fraction equals sin squared theta open parentheses fraction numerator sin squared theta over denominator a end fraction close parentheses cubed plus cos squared theta open parentheses fraction numerator cos squared theta over denominator b end fraction close parentheses cubed equals s i n squared theta open parentheses fraction numerator 1 over denominator a plus b end fraction close parentheses cubed plus cos squared theta open parentheses fraction numerator 1 over denominator a plus b end fraction close parentheses cubed equals open parentheses fraction numerator 1 over denominator a plus b end fraction close parentheses cubed open parentheses s i n squared theta plus c o s squared theta close parentheses equals open parentheses fraction numerator 1 over denominator a plus b end fraction close parentheses cubed cross times 1 equals open parentheses fraction numerator 1 over denominator a plus b end fraction close parentheses cubed equals R H S space left parenthesis P r o v e d right parenthesis

Answered by satyajit samal | 27th Aug, 2015, 11:11: AM

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