Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Prove the following:

sin space left parenthesis straight n plus 1 right parenthesis straight x space sin left parenthesis straight n plus 2 right parenthesis straight x plus cos space left parenthesis straight n plus 1 right parenthesis straight x space cos left parenthesis straight n plus 2 right parenthesis straight x space equals space cosx


Answer

Let (n+2)x = A    and   (n+1)x = B
     L.H.S.  =  sin(n+1)x sin(n+2)x + cos(n+1)x cos(n+2)x
                = sinB sinA + cosB cosA = cosA cosB + sinA sinB
                = cos(A-B) = cos[(n+2)x - (n+1)x]
                = cos (nx + 2x - nx - x) = cosx = R.H.S.
∴          L.H.S. = R.H.S.
Hence, space space sin left parenthesis straight n plus 1 right parenthesis straight x space sin left parenthesis straight n plus 2 right parenthesis straight x space plus space cos left parenthesis straight n plus 1 right parenthesis straight x space cos left parenthesis straight n plus 2 right parenthesis straight x space equals space cosx

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