Principle of Mathematical Induction

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Question

Prove the following:

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

Solution

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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Principle Of Mathematical Induction

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