The problem involves the equation cos(θ)^x + sin(θ)^x = 1, where the goal is to determine the number of real values of x that satisfy this equation for all values of θ.
The discussion is ongoing, with participants exploring different approaches to prove the uniqueness of the solution. There is an acknowledgment of the need to demonstrate the solution for a particular value of θ as a means to support the broader claim.
Participants are working under the assumption that they need to prove the solution holds for all values of θ, and there is a focus on finding a suitable θ to simplify the proof process.
You want to prove there is no x, other than 2, which gives equality for all values of θ.vijayramakrishnan said:but i don't know how to prove that it is the only solution available
thank you for replying sir,i want to prove that 2 is the only solution possible.Nathanael said:You want to prove there is no x, other than 2, which gives equality for all values of θ.
Choose any particular θ and show that the only x-value for that particular θ which gives equality is x=2 (that would prove it, right?)
What particular value of θ might it be easy to show this for?
When you say "the only solution possible," what you really mean is, "the only solution possible for all values of θ" right?vijayramakrishnan said:thank you for replying sir,i want to prove that 2 is the only solution possible.
Nathanael said:When you say "the only solution possible," what you really mean is, "the only solution possible for all values of θ" right?
If you were to choose a particular value of θ and prove that 2 is the only solution for that θ, then that would prove what you want, because if x is not 2 then x does not solve it for that particular θ you chose.
This means you only have to prove it for any single value of θ. Do you agree?
I didn't want to hint at it too much, but if you choose the θ such that cosθ=sinθ then you can factor together the exponent x and complete the proof.