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Question Number 155724 by mathdanisur last updated on 03/Oct/21

$$\mathrm{Find}:$$$$\frac{1}{{\sin }^2{6}^{°}}+\frac{1}{{\sin }^{2}42°}+\frac{1}{{\sin }^{2}66°}+\frac{1}{{\sin }^{2}78°}=?$$

Answered by som(math1967) last updated on 04/Oct/21

$$\frac{1}{{sin}^{2}6}+\frac{1}{{sin}^{2}66}$$$$\frac{2{sin}^{2}66+2{sin}^{2}6}{2{sin}^{2}6{sin}^{2}66}$$$$=\frac{1-cos132+1-cos12}{2{sin}^{2}6{sin}^{2}66}$$$$=\frac{2+cos48-cos12}{2{sin}^{2}6{sin}^{2}66}$$$$=\frac{2+2sin30sin(-18)}{2{sin}^{2}6{sin}^{2}66}$$$$=\frac{2(1-\frac{\sqrt{5}-1}{8})}{2{sin}^{2}6{sin}^{2}66}$$$$=\frac{9-\sqrt{5}}{2{\left(2sin6sin66\right)}^2}$$$$=\frac{9-\sqrt{5}}{{(cos60-cos72)}^2}$$$$=\frac{9-\sqrt{5}}{2{(\frac{1}{2}-\frac{\sqrt{5}-1}{4})}^2}=\frac{8(9-\sqrt{5})}{{(3-\sqrt{5})}^2}=\frac{8(9-\sqrt{5})}{14-6\sqrt{5}}$$$$=\frac{4(9-\sqrt{5})}{7-3\sqrt{5}}=\frac{4(9-\sqrt{5})(7+3\sqrt{5})}{7-3\sqrt{5}}$$$$again$$$$\frac{1}{{sin}^{2}42}+\frac{1}{{sin}^{2}78}$$$$\frac{1-cos84+1-cos156}{2{sin}^{2}42{sin}^{2}78}$$$$=\frac{2+cos24-cos84}{2{sin}^{2}42{sin}^{2}78}$$$$\frac{2+2sin54sin30}{2{sin}^{2}42{sin}^{2}78}$$$$=\frac{2(1+\frac{\sqrt{5}+1}{8})}{2{sin}^{2}42{sin}^{2}78}$$$$=\frac{9+\sqrt{5}}{2{\left(2sin42sin78\right)}^2}=\frac{9+\sqrt{5}}{2{(cos36-cos120)}^2}$$$$=\frac{9+\sqrt{5}}{2{(\frac{\sqrt{5}+1}{4}+\frac{1}{2})}^2}$$$$=\frac{8(9+\sqrt{5})}{{(3+\sqrt{5})}^2}=\frac{8(9+\sqrt{5})}{14+6\sqrt{5}}=\frac{4(9+\sqrt{5})(7-3\sqrt{5})}{(7+3\sqrt{5})(7-3\sqrt{5})}$$$$\therefore \frac{1}{{\sin }^2{6}^{°}}+\frac{1}{{\sin }^{2}42°}+\frac{1}{{\sin }^{2}66°}+\frac{1}{{\sin }^{2}78°}$$$$=\frac{4}{4}(63-15+9\sqrt{5}+63-15-9\sqrt{5})$$$$=126-30=96ans$$$$$$$$$$

Commented by peter frank last updated on 04/Oct/21

$$\mathrm{thanks}$$

Commented by Tawa11 last updated on 04/Oct/21

$$\mathrm{Weldone}\mathrm{sir}.$$

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