The maximum value of $$cos2$$⁡$$45$$∘$$+x+($$$$sin$$⁡x−$$cos$$⁡$$x)2$$ is

$$cos2$$⁡$$45$$∘$$+x+($$$$sin$$⁡x−$$cos$$⁡$$x)2$$ $$=$$$$cos$$⁡$$x2$$−$$sin$$⁡$$x22+($$$$sin$$⁡x−$$cos$$⁡$$x)2$$ $$=32(1$$−$$sin$$⁡$$2x)Hence$$, Maximum value $$=32(1$$−$$($$−$$1))=3$$

Extreme values and Periodicity of Trigonometric functions

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Question

The maximum value of $\cos^{2} (45^{\circ} + x) + (\sin x - \cos x)^{2}$ is

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Solution

$\begin{array}{l} \cos^{2} (45^{\circ} + x) + (\sin x - \cos x)^{2} \\ = {[\frac{\cos x}{\sqrt{2}} - \frac{\sin x}{\sqrt{2}}]}^{2} + (\sin x - \cos x)^{2} \\ = \frac{3}{2} (1 - \sin 2 x) \end{array}$
Hence, Maximum value $= \frac{3}{2} (1 - (- 1)) = 3$

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