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Question
If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.
[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]
Solution
Given that: tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`
⇒ tanθ = `(tanalpha - 1)/(tan alpha + 1)`
= `(tanalpha - tan pi/4)/(1 + tan pi/4 tan alpha)`
⇒ tanθ = `tan(alpha - pi/4)`
∴ θ = `alpha - pi/4`
⇒ cosθ = `cos(alpha - pi/4)`
⇒ cosθ = `cos alpha cos pi/4 + sin alpha sin pi/4`
⇒ cosθ = `cos alpha . 1/sqrt(2) + sin alpha . 1/sqrt(2)`
⇒ `sqrt(2) cos theta` = cosα + sinα
⇒ sinα + cosα = `sqrt(2) cos theta`
Hence proved.
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