1.If cosalpha=1/2(x+1/x) and cosbeta=1/2(y+1/y) then evaluate:
cos(alpha minus beta)=??
2.Find the number of real solutions of the equation cos to the power of 7 x plus sin to the power of 4 x equals 1 in the interval [negative straight pi comma straight pi]?
Please help in solving them...
 

Asked by rakshitreshu1997 | 19th Jul, 2015, 12:45: PM

Expert Answer:

1. C o n s i d e r space t h e space g i v e n space e x p r e s s i o n s colon cos alpha equals 1 half open parentheses x plus 1 over x close parentheses... left parenthesis 1 right parenthesis a n d cos beta equals 1 half open parentheses y plus 1 over y close parentheses... left parenthesis 2 right parenthesis W e space n e e d space t o space f i n d space cos open parentheses alpha minus beta close parentheses W e space k n o w space t h a t comma c o s open parentheses alpha minus beta close parentheses equals cos alpha cross times cos beta plus sin alpha cross times sin beta... left parenthesis 3 right parenthesis W e space h a v e space c o s alpha equals 1 half open parentheses x plus 1 over x close parentheses S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma cos squared alpha equals 1 fourth open parentheses x plus 1 over x close parentheses squared T h u s comma 1 minus c o s squared alpha equals 1 minus 1 fourth open parentheses x plus 1 over x close parentheses squared rightwards double arrow sin squared alpha equals fraction numerator 4 minus open parentheses x squared plus begin display style 1 over x squared end style plus 2 close parentheses over denominator 4 end fraction rightwards double arrow s i n squared alpha equals fraction numerator 2 minus open parentheses x squared plus begin display style 1 over x squared end style close parentheses over denominator 4 end fraction rightwards double arrow s i n alpha equals fraction numerator square root of 2 minus open parentheses x squared plus begin display style 1 over x squared end style close parentheses end root over denominator 2 end fraction S i m i l a r l y comma space sin beta equals fraction numerator square root of 2 minus open parentheses y squared plus begin display style 1 over y squared end style close parentheses end root over denominator 2 end fraction therefore sin alpha cross times sin beta equals fraction numerator square root of 2 minus open parentheses x squared plus begin display style 1 over x squared end style close parentheses end root over denominator 2 end fraction cross times fraction numerator square root of 2 minus open parentheses y squared plus begin display style 1 over y squared end style close parentheses end root over denominator 2 end fraction rightwards double arrow s i n alpha cross times s i n beta equals 1 fourth cross times square root of 2 minus open parentheses x squared plus 1 over x squared close parentheses end root cross times square root of 2 minus open parentheses y squared plus 1 over y squared close parentheses end root rightwards double arrow s i n alpha cross times s i n beta equals 1 fourth cross times square root of open square brackets 2 minus open parentheses x squared plus 1 over x squared close parentheses close square brackets open square brackets 2 minus open parentheses y squared plus 1 over y squared close parentheses close square brackets end root rightwards double arrow s i n alpha cross times s i n beta equals 1 fourth cross times square root of open square brackets open parentheses x squared plus 1 over x squared close parentheses minus 2 close square brackets open square brackets open parentheses y squared plus 1 over y squared close parentheses minus 2 close square brackets end root rightwards double arrow s i n alpha cross times s i n beta equals 1 fourth cross times square root of open parentheses x minus 1 over x close parentheses squared open parentheses y minus 1 over y close parentheses squared end root rightwards double arrow s i n alpha cross times s i n beta equals 1 fourth cross times open parentheses x minus 1 over x close parentheses open parentheses y minus 1 over y close parentheses... left parenthesis 4 right parenthesis A n d space cos alpha cross times cos beta equals 1 half open parentheses x plus 1 over x close parentheses cross times 1 half open parentheses y plus 1 over y close parentheses space... left square bracket f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis right square bracket rightwards double arrow space c o s alpha cross times c o s beta equals 1 fourth open parentheses x plus 1 over x close parentheses open parentheses y plus 1 over y close parentheses... left parenthesis 5 right parenthesis F r o m space e q u a t i o n s space left parenthesis 3 right parenthesis comma space left parenthesis 4 right parenthesis space a n d space left parenthesis 5 right parenthesis comma space w e space h a v e comma c o s open parentheses alpha minus beta close parentheses equals 1 fourth open parentheses x plus 1 over x close parentheses open parentheses y plus 1 over y close parentheses minus 1 fourth open parentheses x minus 1 over x close parentheses open parentheses y minus 1 over y close parentheses rightwards double arrow c o s open parentheses alpha minus beta close parentheses equals 1 fourth open square brackets open parentheses x plus 1 over x close parentheses open parentheses y plus 1 over y close parentheses minus open parentheses x minus 1 over x close parentheses open parentheses y minus 1 over y close parentheses close square brackets
Please post question 2 as a separate question.

Answered by Vimala Ramamurthy | 20th Jul, 2015, 08:49: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.