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Question

Which of the following statement is/are correct for $$0 < \theta < \dfrac{\pi}{2}$$?


A
(cosθ)1/2cosθ2
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B
(cosθ)3/4cos3θ4
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C
cos5θ6(cosθ)5/6
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D
cos7θ8(cosθ)7/8
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Solution

The correct options are
A $$(\cos \theta)^{1/2} \le \cos\dfrac{\theta}{2}$$
D $$\cos\dfrac{5 \theta}{6} \ge (\cos \theta)^{5/6}$$
$$0<\theta<\dfrac{\pi }{2}$$
$$\Rightarrow 0<\dfrac{\theta}{n}<\dfrac{\pi }{2n}$$

In the range $$(0,\dfrac{\pi }{2}),$$  $$cos(\dfrac{\theta}{n} )$$ ranges from $$1$$ to $$\text{some positive value}$$ whereas $$cos\theta$$ ranges from $$1$$ to $$0$$
So,  $$cos(\dfrac{\theta}{n})>cos(\theta)\Rightarrow cos(\dfrac{\theta}{n})>(cos\theta)^{1/n}$$
So, Options (A),(C)

Mathematics

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