Showing that if x = \sec\theta + \tan\theta ,then x + \frac{1}{x} = 2\sec\theta I us

Maximillian Patterson

Maximillian Patterson

Answered question

2022-05-02

Showing that if x=secθ+tanθ ,then x+1x=2secθ
I used a few simple trig identities but get nowhere. I am confused about this question a lot. I do not see where the theta and x can be in one.
None are squared so I cannot used the identities I would think. The only thing I can think to do is x=1sinθ+cosθsinθ

Answer & Explanation

Sergio Kidd

Sergio Kidd

Beginner2022-05-03Added 14 answers

Your first problem is that
sec{θ}=1cos{θ},tan{θ}=sin{θ}cos{θ} ,
so actually
x=1cos{θ}+sin{θ}cos{θ}=1+sin{θ}cos{θ}
Then
x+1x=1+sin{θ}cos{θ}+cos{θ}1+sin{θ}
=(1+sin{θ})2+cos2{θ}cos{θ}(1+sin{θ})
=1+2sin{θ}+(sin2{θ}+cos2{θ})cos{θ}(1+sin{θ})
=2(1+sin{θ})cos{θ}(1+sin{θ})=2sec{θ}
so in fact the identity is not true.
aimadorsozf

aimadorsozf

Beginner2022-05-04Added 10 answers

Here is a detailed answer.
secθ+tanθ+1secθ+tanθ=secθsecθ+tanθ+tanθsecθ+tanθ+1secθ+tanθ
=secθ+tanθsecθ+tanθ+1secθ+tanθ
=secθ+tanθsecθ+tanθ+sec2θ-tan2θsecθ+tanθ
=sec2θ+2secθtanθ+tan2θ+sec2θ-tan2θsecθ+tanθ
=2sec2θ+2secθtanθsecθ+tanθ
=2secθsecθ+tanθsecθ+tanθ
=2secθ
We substituted 1 with sec2θ-tan2θ

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