If-f-x-sin-2-x-sin-2-x-pi-3-cos-x-cos-x-pi-3-and-g-5-4-1-then-gof-x-

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Question Number 248 by sushmitak last updated on 25/Jan/15

$$\mathrm{If}f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\pi /3)+\cos x\cos (x+\pi /3)$$$$\mathrm{and}g\left(5/4\right)=1\mathrm{then}gof\left(x\right)=?$$

Answered by 123456 last updated on 17/Dec/14

$$f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$$$$\cos (x+\frac{\pi }{3})=\cos x\cos \frac{\pi }{3}-\sin x\sin \frac{\pi }{3}$$$$=\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x$$$$\cos x\cos (x+\frac{\pi }{3})=\frac{1}{2}{\cos }^{2}x-\frac{\sqrt{3}}{2}\cos x\sin x$$$$\sin (x+\frac{\pi }{3})=\sin x\cos \frac{\pi }{3}+\cos x\sin \frac{\pi }{3}$$$$=\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x$$$${\sin }^2(x+\frac{\pi }{3})=\frac{1}{4}{\sin }^{2}x+\frac{\sqrt{3}}{2}\cos x\sin x+\frac{3}{4}\cos {}^{2}x$$$$f\left(x\right)={\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos x\cos (x+\frac{\pi }{3})$$$$={\sin }^{2}x+\frac{1}{4}{\sin }^{2}x+\frac{\sqrt{3}}{2}\cos x\sin x+\frac{3}{4}{\cos }^{2}x+\frac{1}{2}{\cos }^{2}x-\frac{\sqrt{3}}{2}\cos x\sin x$$$$=(1+\frac{1}{4}){\sin }^{2}x+(\frac{3}{4}+\frac{1}{2}){\cos }^{2}x$$$$=\frac{5}{4}{\sin }^{2}x+\frac{5}{4}{\cos }^{2}x$$$$=\frac{5}{4}({\sin }^2\mathit{x}+{\cos }^2\mathit{x})=\frac{5}{4}$$$$\mathbf{then}$$$$g\left[f\left(x\right)\right]=g\left(\frac{5}{4}\right)=1$$

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