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Question Number 100695 by john santu last updated on 28/Jun/20

Commented by bobhans last updated on 28/Jun/20

$$\sin 18=\frac{\sqrt{5}-1}{4},\sin 54=\cos 36=1-2\sin {}^{2}18$$$$\sin 54=1-2\left(\frac{6-2\sqrt{5}}{16}\right)=1-\left(\frac{3-\sqrt{5}}{4}\right)=\frac{1+\sqrt{5}}{4}$$$$\Rightarrow \frac{1}{\sin 18}-\frac{1}{\sin 54}=\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}$$$$=4\left(\frac{\sqrt{5}+1-\sqrt{5}+1}{4}\right)=2$$

Commented by john santu last updated on 28/Jun/20

$$\mathrm{correct}\mathrm{sir}\mathrm{bob}$$

Commented by Dwaipayan Shikari last updated on 28/Jun/20

$$(\frac{1}{\sqrt{5}-1}-\frac{1}{\sqrt{5}+1})4=2$$

Answered by EquationMaker2305 last updated on 28/Jun/20

$$0.45800357$$

Commented by prakash jain last updated on 28/Jun/20

$$\mathrm{You}\mathrm{seem}\mathrm{to}\mathrm{have}\mathrm{calculated}\mathrm{this}$$$$\mathrm{value}\mathrm{using}\mathrm{a}\mathrm{calculator}\mathrm{in}\mathrm{radian}$$$$\mathrm{mode}.\mathrm{Change}\mathrm{mode}\mathrm{ro}\mathrm{degrees}$$$$\mathrm{and}\mathrm{you}\mathrm{will}\mathrm{get}\mathrm{correct}\mathrm{answer}.$$

Commented by EquationMaker2305 last updated on 28/Jun/20

Please post plain text comments instead, it much readable.

Answered by 1549442205 last updated on 28/Jun/20

$$\mathrm{We}\mathrm{have}\cos 54°=\sin 36°\Rightarrow {4\cos }^{3}18°-3\cos 18°=2\sin 18°\cos 18°$$$$\iff {4\cos }^{2}18°-3=2\sin 18°\iff 4(1-{\sin }^{2}18°)-3=2\sin 18°$$$$\iff {4\sin }^{2}18°+2\sin 18°-1=0.\mathrm{We}\mathrm{look}\mathrm{at}\mathrm{it}\mathrm{like}$$$$\mathrm{as}\mathrm{a}\mathrm{quadratic}\mathrm{equation}\mathrm{with}\mathrm{respect}\sin 18°.$$$$\mathrm{Then}{\mathrm{\Delta }}^{\prime }=1+4=5.\mathrm{Hence},\sin 18°=\frac{-1+\sqrt{5}}{4}$$$$\mathrm{and}\mathrm{so}\sin 54°=\cos 36°=1-{2\sin }^{2}18°=$$$$1-2\times \frac{6-2\sqrt{5}}{16}=1-\frac{3-\sqrt{5}}{4}=\frac{\sqrt{5}+1}{4},\mathrm{so}$$$$\frac{1}{\mathbf{sin}18°}-\frac{1}{\mathbf{sin}54°}=\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}=2$$$$$$$$$$

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