Multiple Choice Questions

Consider the following statements 1 cos sec can never be equal to 1 5 2 sec2 cos...........

Quantitative Aptitude (CDS) Trignometry

Consider the following statements :

1. cos $\Theta$ + sec $\Theta$ can never be equal to 1.5.

2. sec² $\Theta$ + cosec² $\Theta$ can never be less than 4.

Which of the statements given above is/are correct?

1 only

2 only

Both 1 and 2

Neither 1 nor 2

(1) As we know that Am $\geq$ Gm

$\frac{a+b}{2}\: \geq \: \sqrt{ab}$

$\frac{cos\Theta\:+ sec\Theta }{2}\geq 2\sqrt{1}$

$\frac{cos\Theta\:+ sec\Theta }{2}\geq 2$ hence (1) is true

$2)\;sec^{2}\Theta +cosec^{2}\Theta$

$1+tan^{2}\Theta +1+cot^{2}\Theta$

$2+tan^{2}\Theta +cot^{2}\Theta ---------(1)$

As we know

$tan^{2}\Theta +cot^{2}\Theta \geq 2$

using equation (1) we have

$2+tan^{2}\Theta +cot^{2}\Theta \geq 2+2$

$2+tan^{2}\Theta +cot^{2}\Theta \geq 4$

Hence (2) is also correct

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