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Formation of a Differential Equation from a General Solution

Solve the fol...

Question

Solve the following equations.
Find all the solutions of the equation $s i n (4 x + \frac{π}{4}) + c o s (4 x + \frac{5 π}{4}) = \sqrt{2}$ which satisfy the inequality $\frac{c o s 2 x}{c o s 2 - s i n 2} > 2 - s i n 4 x .$

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Solution

Given
$s i n (4 x + \frac{π}{4}) + c o s (4 x \frac{5 π}{4}) = \sqrt{2}$
Let us now consider L.H.S
L.H.S $= s i n [4 x + \frac{π}{4}] + c o s [4 x + \frac{5 π}{4}] \to e q (1)$
as we know that $s i n (A + B) = s i n A c o s B + c o s A s i n B$
$c o s (A + B) = c o s A c o s B - s i n A s i n B$
Let us now apply the above froms formation to eq(1)
$\Leftrightarrow s i n 4 x c o s \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + c o s 4 x c o s \frac{5 π}{4} - s i n 4 x s i n \frac{5 π}{4}$
$\Leftrightarrow s i n 4 x c o s \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + c o s x 4 c o s [π \frac{π}{4}] - s i n 4 x s i n [π + \frac{π}{4}]$
$[∵ 180 + θ$ is in $Q I I I$ $c o s + v e s i n - v e]$
$\Leftrightarrow s i n 4 x c o s \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + c o s 4 x c o s \frac{π}{4} + s i n 4 x s i n \frac{π}{4}$
$[∵ s i n (90 - θ) = c o s θ & c o s (90 - θ) = s i n θ]$
$s i n 4 x c o s \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + c o s 4 x c o s (\frac{π}{2} - \frac{π}{4}) + s i n 4 x s i n (\frac{π}{2} - \frac{π}{4})$
$\Leftrightarrow s i n 4 x c o s \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + c o s 4 x s i n \frac{π}{4} + s i n 4 x c o s \frac{π}{4}$
$(∵ s i n \frac{π}{4} = c o s \frac{π}{4} = 1 / \sqrt{2})$
$\Leftrightarrow 2 s i n 4 x [\frac{1}{\sqrt{2}}] + 2 c o s 4 x [\frac{1}{\sqrt{2}}]$
$[∵ 2 = \sqrt{2} . \sqrt{2}]$
$\Rightarrow \frac{2}{\sqrt{2}} [s i n 4 x + c o s 4 x] \Leftrightarrow \sqrt{2} [s i n 4 x + c o s 4 x] \to e q (2)$
$[∵ s i n 2 A = 2 s i n c o s A, c o s 2 A = c o s^{2} A - s i n^{2}]$
Let as apply the above transformation to eq(2)
$\Leftrightarrow \sqrt{2} [s i n 2 (2 x) + c o s 2 (2 x)]$
$\Leftrightarrow \sqrt{2} [2 s i n 2 x c o s 2 x + c o s^{2} 2 x - s i n^{2} 2 x]$
$[∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b]$
$\Leftrightarrow \sqrt{2} [2 s i n 2 x c o s 2 x + c o s^{2} 2 x - s i n^{2} 2 x]$
$\Leftrightarrow \sqrt{2} [(s i n 2 x + c o s 2 x)^{2}]$
$\Leftrightarrow \sqrt{2} [(0 - 1)^{2}] \Rightarrow \sqrt{2} (- 1)^{2}$
[ x is considered as $180^{\circ}$ & as both the fuction ile in Q]
$[∵ s i n 180^{\circ} = 0 c o s 180^{\circ} = - 1]$
$\Rightarrow \sqrt{2} \Rightarrow$ R.H.S
$∴$ L.H.S = R.H.S

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