Given that θ = (\pi)/(4) , which of the following shows that cosθsecθ = 1? - ((√(2) )/(2))(√(2)) - ((√(3) )/(2))(2(√(3))/(3) ) - (\frac{{1} }{2})(2)

Question

Given that θ =
`(\pi)/(4)` , which of the following shows that cosθsecθ = 1?

-
`((√(2) )/(2))(√(2))`
-
`((√(3) )/(2))(2(√(3))/(3) )`
-
`(\frac{{1} }{2})(2)`

Answer 1

Option 1.

Explanation:

It is given that

`\theta=(\pi)/(4)`

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

We know that,

`\cos \left( (\pi)/(4)\right)=(1)/(√(2))`

On rationalization, we get

`\cos \left( (\pi)/(4)\right)=(1)/(√(2))* (√(2))/(√(2))=(√(2))/(2)`

`\sec \left( (\pi)/(4)\right)=√(2)`

Now,

`LHS=\cos \theta\sec \theta`

`LHS=\left((√(2))/(2)\right)(√(2))`

Therefore, the correct option is 1.

Answer 2

`cos((\pi )/(4)) = (√(2))/(2)` per Unit Circle

`sec((\pi )/(4)) = (1 )/(cos((\pi )/(4))) = (1)/((√(2))/(2)) = (2)/(√(2)) = (2√(2))/(2)` =
`√(2)`

cosθ * secθ = 1

`(√(2) )/(2)` *
`√(2)` = 1

1 = 1

Answer: A