Question

Given that θ = i}{4}" align="absmiddle" class="latex-formula"> , which of the following shows that cosθsecθ = 1?
- $(\frac{\sqrt{2} }{2})(\sqrt{2})$
- $(\frac{\sqrt{3} }{2})(2\frac{\sqrt{3}}{3} )$
- $(\frac{{1} }{2})(2)$

Answer 1

$cos(\frac{\pi }{4}) = \frac{\sqrt{2}}{2}$   per Unit Circle

$sec(\frac{\pi }{4}) = \frac{1 }{cos(\frac{\pi }{4})} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2}$ = $\sqrt{2}$

cosθ * secθ = 1

$\frac{\sqrt{2} }{2}$ * $\sqrt{2}$ = 1

               1 = 1

Answer: A

Answer 2

Answer:

Option 1.

Step-by-step explanation:

It is given that

$\theta=\dfrac{\pi}{4}$

We need to find the expression that shows $\cos \theta\sec \theta=1$.

We know that,

$\cos \left( \dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}$

On rationalization, we get

$\cos \left( \dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}$

$\sec \left( \dfrac{\pi}{4}\right)=\sqrt{2}$

Now,

$LHS=\cos \theta\sec \theta$

$LHS=\left(\dfrac{\sqrt{2}}{2}\right)(\sqrt{2})$

Therefore, the correct option is 1.