The range of $$cos$$θ $$($$$$sin$$θ $$+sin2$$θ$$+sin2$$α$$)$$ is

Question

The range of $$cos$$θ $$($$$$sin$$θ $$+sin2$$θ$$+sin2$$α$$)$$ is

Infinity Learn · Accepted Answer

Let  fθ$$=$$$$cos$$ θ $$($$$$sin$$ θ $$+sin2$$θ$$+sin2$$α$$)$$⇒$$f($$θ$$)$$$$sec$$θ−$$sin$$θ$$2=sin2$$θ$$+sin2$$α⇒$$f2($$θ$$)1+tan2$$θ−$$2f($$θ$$)$$$$tan$$θ−$$sin2$$α$$=0$$⇒$$f2($$θ$$)tan2$$θ−$$2f($$θ$$)$$$$tan$$θ$$+(f2($$θ$$)$$−$$sin2$$α$$)=0$$∵  $$tan$$θ  is real,    we  have  $$b2$$−$$4ac$$≥$$0$$⇒$$4f2($$θ$$)$$−$$4f2($$θ$$)(f2($$θ$$)$$−$$sin2$$α$$)$$≥$$0$$⇒$$4f2($$θ$$)1$$−$$f2($$θ$$)+sin2$$α≥$$0$$⇒$$1$$−$$f2($$θ$$)+sin2$$α≥$$0$$⇒$$f2($$θ$$)$$≤$$1+sin2$$α⇒−$$1+sin2$$α≤$$f($$θ$$)$$≤$$1+sin2$$α

The range of $\cos θ (\sin θ + \sqrt{\sin^{2} θ + \sin^{2} α})$ is

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The range of cosθ (sinθ +sin2θ+sin2α) is

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The range of $\cos θ (\sin θ + \sqrt{\sin^{2} θ + \sin^{2} α})$ is