Prove that:
(i) cos(A-B)= cosAcosB + sinAsinB
(ii)cos(A+B)= cosAcosB - sinAsinB

Asked by sunil2791 | 14th Nov, 2017, 07:46: PM

Expert Answer:

begin mathsize 16px style We space know space that space equal space chords space of space straight a space circle space makes space equal space angles
at space the space centre space and space chords space straight P subscript 0 straight P subscript 3 space and space straight P subscript 1 straight P subscript 2 space subtend space equals space
angles space at space the space centre. space
straight P subscript 0 straight P subscript 3 space equals space straight P subscript 1 straight P subscript 2
Using space distance space formula comma
square root of open square brackets cos open parentheses straight A minus straight B close parentheses squared minus 1 close square brackets plus sin squared open parentheses straight A minus straight B close parentheses end root equals square root of open parentheses cosB minus cosA close parentheses squared plus open parentheses sinB minus sinA close parentheses squared end root
After space solving space you space will space get space cos left parenthesis straight A minus straight B right parenthesis equals cosAcosB plus sinAsinB

For space the space second space one space we space can space solve space using space above space formula
cos left parenthesis straight A plus straight B right parenthesis equals cos open square brackets straight A minus left parenthesis negative straight B right parenthesis close square brackets
cos left parenthesis straight A plus straight B right parenthesis equals cosAcos left parenthesis negative straight B right parenthesis plus sinAsin left parenthesis negative straight B right parenthesis
cos left parenthesis straight A plus straight B right parenthesis equals cosAcosB minus sinAsinB space space space space space space space space space space space space space space space space cos left parenthesis negative straight A right parenthesis equals cosA space and space sin left parenthesis negative straight B right parenthesis equals negative sinB
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Answered by Sneha shidid | 7th Dec, 2017, 10:20: AM