If θ+ϕ=π2where θ and ϕare positive, then (sin⁡θ+sin⁡ϕ)sin⁡π4is always less than ______

If M is mid point of PQ, then M=θ+ϕ2,sin⁡θ+sin⁡ϕ2Also, N≡θ+ϕ2,sin⁡θ+ϕ2It is clear from the figure ML ≤ NL ⇒sin⁡θ+sin⁡ϕ2≤sin⁡θ+ϕ2⇒sin⁡θ+sinϕ≤2sin⁡θ+ϕ2⇒2sin⁡π4=2∴sin⁡θ+sin⁡ϕ≤2 and (sin⁡θ+sin⁡ϕ)sin⁡π4≤1

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If $θ + ϕ = \frac{π}{2}$ where $θ and ϕ$ are positive, then $(\sin θ + \sin ϕ) \sin (\frac{π}{4})$ is always less than ______

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Detailed Solution

If M is mid point of PQ, then $M = (\frac{θ + ϕ}{2}, \frac{\sin θ + \sin ϕ}{2})$
Also, $N \equiv (\frac{θ + ϕ}{2}, \sin (\frac{θ + ϕ}{2}))$
It is clear from the figure ML $\leq$ NL

$\begin{array}{l} \Rightarrow \frac{\sin θ + \sin ϕ}{2} \leq \sin (\frac{θ + ϕ}{2}) \\ \Rightarrow \sin θ + sinϕ \leq 2 \sin (\frac{θ + ϕ}{2}) \\ \Rightarrow 2 \sin (\frac{π}{4}) = \sqrt{2} \\ ∴ \sin θ + \sin ϕ \leq \sqrt{2} and \\ (\sin θ + \sin ϕ) \sin \frac{π}{4} \leq 1 \end{array}$