Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

space space space cos open parentheses straight pi over 4 minus straight x close parentheses space cos open parentheses straight pi over 4 minus straight y close parentheses minus sin open parentheses straight pi over 4 minus straight x close parentheses space sin open parentheses straight pi over 4 minus straight y close parentheses space equals space sin left parenthesis straight x plus straight y right parenthesis




Answer

Let open parentheses straight pi over 4 minus straight x close parentheses space equals space straight A and straight pi over 4 minus straight y space equals space straight B
                    L.H.S. = cos open parentheses straight pi over 4 minus straight x close parentheses space cos open parentheses straight pi over 4 minus straight y close parentheses minus sin open parentheses straight pi over 4 minus straight x close parentheses space sin open parentheses straight pi over 4 minus straight y close parentheses

                             equals space cosA space cosB space minus space sinA space sinB space equals space cos left parenthesis straight A plus straight B right parenthesis
                             equals cos open square brackets open parentheses straight pi over 4 minus straight x close parentheses plus open parentheses straight pi over 4 minus straight y close parentheses close square brackets space equals space cos open square brackets straight pi over 2 minus left parenthesis straight x plus straight y right parenthesis close square brackets
                              equals sin left parenthesis straight x plus straight y right parenthesis                    open square brackets because space cos open parentheses straight pi over 2 minus straight theta close parentheses space equals space sinθ close square brackets
                              = R.H.S.
∴                   L.H.S. = R.H.S
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