Let $$y=$$$$cos$$⁡x−$$sin$$⁡$$x(y$$>$$0)$$ $$y2=1$$−$$2sin$$⁡x⋅$$cos$$⁡x $$cos$$⁡$$x+$$$$sin$$⁡$$x=54$$∴ $$1+2sin$$⁡xcos⁡$$x=2516or$$ $$2sin$$⁡x⋅$$cos$$⁡$$x=916Now$$, $$y2=1$$−$$2sin$$⁡xcos⁡$$x=1$$−$$916=716$$

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Question

If $0 < x < \frac{π}{4}$ and $\cos x + \sin x = \frac{5}{4}$ then the value of $16 (\cos x - \sin x)^{2}$ is

Solution

$\begin{array}{l} Let y = \cos x - \sin x (y > 0) \\ y^{2} = 1 - 2 \sin x \cdot \cos x \\ \cos x + \sin x = \frac{5}{4} \\ ∴ 1 + 2 \sin xcos x = \frac{25}{16} \end{array}$
or $2 \sin x \cdot \cos x = \frac{9}{16}$
Now, $y^{2} = 1 - 2 \sin xcos x = 1 - \frac{9}{16} = \frac{7}{16}$

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