If $$sin$$−$$1$$⁡$$x+$$$$sin$$−$$1$$⁡$$y+$$$$sin$$−$$1$$⁡$$z=3$$$$π$$$$2$$ and $$f(1)$$ $$=$$ $$1$$, $$f(p$$ $$+$$ $$q)$$ $$=$$ $$f(p)$$. $$f(q)$$∀p, q ∈ R then, $$xf(1)+y(2)+zf(3)$$−$$x+y+zxf(1)+yf(2)+zf(3)=$$

Question

If $$sin$$−$$1$$⁡$$x+$$$$sin$$−$$1$$⁡$$y+$$$$sin$$−$$1$$⁡$$z=3$$$$π$$$$2$$ and $$f(1)$$ $$=$$ $$1$$, $$f(p$$ $$+$$ $$q)$$ $$=$$ $$f(p)$$. $$f(q)$$∀p, q ∈ R then, $$xf(1)+y(2)+zf(3)$$−$$x+y+zxf(1)+yf(2)+zf(3)=$$

Infinity Learn · Accepted Answer

Since, −π$$2$$≤$$sin$$−$$1$$⁡x≤π$$2$$∴$$sin$$−$$1$$⁡$$x+$$$$sin$$−$$1$$⁡$$y+$$$$sin$$−$$1$$⁡$$z=3$$$$π$$$$2$$⇒$$sin$$−$$1$$⁡$$x=$$$$sin$$−$$1$$⁡$$y=$$$$sin$$−$$1$$⁡$$z=$$π$$2$$⇒$$x=y=z=1Also$$, $$f(p$$ $$+$$ $$q)$$ $$=$$ $$f(p)$$. $$f(q)$$ ∀ p, q ∈ R      …. (1)Given$$, $$f(1) $$=$$ $$1From$$ (1),$$f(1$$ $$+$$ $$1)$$ $$=$$ $$f(1)$$. $$f(1)$$⇒ $$f(2)$$ $$=12$$ $$=$$ $$1$$               ………. (2)From$$ $$(2), $$f(2$$ $$+$$ $$1)$$ $$=$$ $$f(2)$$ . $$f(1)$$⇒ $$f(3)$$ $$=$$ $$12$$ $$.1$$ $$=$$ $$13$$ $$=$$ $$1Now$$, given expression $$=3$$−$$33=2$$

If $\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}$ and f(1) = 1, f(p + q) = f(p). f(q) $\forall$ p, q ∈ R then, $x^{f (1)} + y^{(2)} + z^{f (3)} - \frac{x + y + z}{x^{f (1)} + y^{f (2)} + z^{f (3)}} =$

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If sin−1⁡x+sin−1⁡y+sin−1⁡z=3π2 and f(1) = 1, f(p + q) = f(p). f(q)∀p, q ∈ R then, xf(1)+y(2)+zf(3)−x+y+zxf(1)+yf(2)+zf(3)=

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If $\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}$ and f(1) = 1, f(p + q) = f(p). f(q) $\forall$ p, q ∈ R then, $x^{f (1)} + y^{(2)} + z^{f (3)} - \frac{x + y + z}{x^{f (1)} + y^{f (2)} + z^{f (3)}} =$