Question

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

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

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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