If $(a^{2} - b^{2}) s i n x - 2 a b c o s x = a^{2} + b^{2}; t h e n t a n x = ?$

(a² + b²)/2ab

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(a² - b²)/2

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(a² - b²)/2ab

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(a² + b²)/2

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(a² + b²)/ab

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Solution

The correct option is C
(a² - b²)/2ab

$\begin{matrix} (a^{2} - b^{2}) s i n x - 2 a b c o s x = a^{2} + b^{2} \Rightarrow a^{2} s i n x - b^{2} s i n x - 2 a b c o s x = a^{2} + b^{2} \Rightarrow a^{2} s i n x - b^{2} s i n x - 2 a b c o s x - a^{2} - b^{2} = 0 \Rightarrow a^{2} (1 - s i n x) + b^{2} (1 + s i n x) + 2 a b c o s x = 0 \Rightarrow {[(a \sqrt{1 - s i n x})]}^{2} + {[b \sqrt{(1 + s i n x)}]}^{2} + 2 a b c o s x = 0 \Rightarrow {[(a \sqrt{1 - s i n x})]}^{2} + {[b \sqrt{(1 + s i n x)}]}^{2} + 2 a b \sqrt{c o s^{2} x} = 0 \Rightarrow {[(a \sqrt{1 - s i n x})]}^{2} + {[b \sqrt{(1 + s i n x)}]}^{2} + 2 a b \sqrt{(1 - s i n^{2} x)} = 0 \Rightarrow {[(a \sqrt{1 - s i n x})]}^{2} + {[b \sqrt{(1 + s i n x)}]}^{2} + 2 a b \sqrt{(1 + s i n x) (1 - s i n x)} = 0 \Rightarrow {[(a \sqrt{1 - s i n x}) + b \sqrt{(1 + s i n x)}]}^{2} = 0 \Rightarrow a \sqrt{1 - s i n x} = - b \sqrt{1 + s i n x} \Rightarrow (a^{2} + b^{2}) s i n x = a^{2} - b^{2} \Rightarrow s i n x = (a^{2} - b^{2}) / (a^{2} + b^{2}) \Rightarrow c o s x = 2 a b / (a^{2} + b^{2}) [c o s x = r o o t {1 - s i n^{2} x}] \Rightarrow t a n x = (a^{2} - b^{2}) / (2 a b) \end{matrix}$

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