Question

sin⁻¹(sinx+cosx/√2) = π/4 + x, - π/4 < x < π/4,Prove it

Answer 1

Answer:

$sin^{-1}(\dfrac{sinx+cosx}{\sqrt{2}})$=$\dfrac{\pi}{4} +x$

Step-by-step explanation:

L.H.S= $sin^{-1}(\dfrac{sinx+ cosx}{\sqrt{2}})\\(\because sin\dfrac{\pi}{4}= \dfrac{1}{\sqrt{2}} =cos\dfrac{\pi}{4})\\=sin^{-1}(sinx\times cos\dfrac{\pi}{4} +cosx\times sin\dfrac{\pi}{4}) \\\because sin(a+b)=sina.cosb +cosa.sinb\ \ \ (formula)\\ =sin^{-1}(sin(\dfrac{\pi}{4}+x))\\\because sin^{-1}(sina)=a \ \ (formula)\\ = \dfrac{\pi}{4} +x =R.H.S$      PROVED