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Definition of Functions

If a cosθ +...

Question

If $a cos θ + b sin θ = m$ and $a sin θ - b cos θ = n$ , then $a^{2} + b^{2}$ equals

A

m + n

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B

m n

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C

\sqrt{m n}

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D

m^{2} + n^{2}

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Solution

The correct option is D $m^{2} + n^{2}$
Squaring $m$ and $n$ and adding them, we get
$m^{2} + n^{2} = (a c o s θ + b s i n θ)^{2} + (a s i n θ - b c o s θ)^{2}$
$= a^{2} c o s^{2} θ + b^{2} s i n^{2} θ + 2 a b s i n θ c o s θ + a^{2} s i n^{2} θ + b^{2} c o s^{2} θ - 2 a b s i n θ c o s θ$
$= a^{2} c o s^{2} θ + b^{2} s i n^{2} θ + a^{2} s i n^{2} θ + b^{2} c o s^{2} θ$
$= a^{2} (c o s^{2} θ + s i n^{2} θ) + b^{2} (c o s^{2} θ + s i n^{2} θ)$
$= a^{2} + b^{2}$

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