Prove that f(x)=sin⁡(x+α)sin⁡(x+β) is monotonic in any interval of its domain, α,β∈RObviously x+β≠πn. Then i've tried to split the problem into three parts: α=β,α&gt;β,β&gt;α.For the first case it's obvious that the function turns into a constant and therefore is monotonic.Since f(x) is periodic we may use that fact and consider some interval for example (0,π2) or (0,π).But since α and β may take any values i didn't manage to figure out any inequality to start from.What steps should i take to proceed?

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Prove that f(x)=sin⁡(x+α)sin⁡(x+β) is monotonic in any interval of its domain, α,β∈RObviously x+β≠πn. Then i&#039;ve tried to split the problem into three parts: α=β,α&amp;gt;β,β&amp;gt;α.For the first case it&#039;s obvious that the function turns into a constant and therefore is monotonic.Since f(x) is periodic we may use that fact and consider some interval for example (0,π2)  or  (0,π).But since α  and  β may take any values i didn&#039;t manage to figure out any inequality to start from.What steps should i take to proceed?

Jacob Trujillo · Accepted Answer

Using  sin⁡((x+β)+(α−β))=sin⁡(x+β)cos⁡(α−β)+cos⁡(x+β)sin⁡(α−β)  one gets  f(x)=cos⁡(α−β)+cot⁡(x+β)sin⁡(α−β)  and the cotangent is decreasing on each interval of its domain.  You could also attack the problem by substituting x+α=y first, then the expression becomes  sin⁡(y+α−β)sin⁡(y)  Now substitute γ=α−β ,and you are left with  g(y)=sin⁡(y+γ)sin⁡(y)  which might be easier to handle.

Prove that f(x) = \frac{\sin(x+\alpha)}{\sin(x+\beta)} is monotonic in any interval of its dom

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