Question

If a cos α + b sin α = m and a sin α - b cos α = n, prove that a2 + b2 = m2 + n2​

Answer 1

Step-by-step explanation:

$acos(\alpha) + bsin(\alpha) = m\\\\(acos(\alpha) + bsin(\alpha))^{2} = m^{2}\\\\similarly\\\\(asin(\alpha) - bcos(\alpha))^{2} = n^{2}\\\\add\\\\a^{2}cos^{2}(\alpha) + b^{2}sin^{2}(\alpha) + 2abcos(\alpha)sin(\alpha) + b^{2}cos^{2}(\alpha) + a^{2}sin^{2}(\alpha) - 2abcos(\alpha)sin(\alpha) = m^{2}+n^{2} \\\\a^{2}(cos^{2}(\alpha) + sin^{2}(\alpha)) + b^{2}(cos^{2}(\alpha) + sin^{2}(\alpha)) = m^{2}+n^{2}\\\\a^{2}+b^{2} = m^{2}+n^{2}$

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