JEE Main 2021 — Mathematics Trigonometry

$$\Rightarrow 2\mathrm{sin}(\frac{\pi }{8})\mathrm{sin}(\frac{2\pi }{8})\mathrm{sin}(\frac{3\pi }{8})\mathrm{sin}(\frac{5\pi }{8})\mathrm{sin}(\frac{6\pi }{8})\mathrm{sin}(\frac{7\pi }{8})$$⇒2sin(8π​)sin(82π​)sin(83π​)sin(85π​)sin(86π​)sin(87π​)

$$\Rightarrow 2\mathrm{sin}(\frac{\pi }{8})\mathrm{sin}(\frac{2\pi }{8})\mathrm{sin}(\frac{3\pi }{8})\mathrm{sin}(\pi -\frac{3\pi }{8})\mathrm{sin}(\pi -\frac{2\pi }{8})\mathrm{sin}(\pi -\frac{\pi }{8})$$⇒2sin(8π​)sin(82π​)sin(83π​)sin(π−83π​)sin(π−82π​)sin(π−8π​)

$$\Rightarrow 2{\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{sin}}^{2}(\frac{2\pi }{8}){\mathrm{sin}}^{2}(\frac{3\pi }{8})$$⇒2sin28π​sin2(82π​)sin2(83π​)

$$\Rightarrow 2{\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{sin}}^{2}(\frac{2\pi }{8}){\mathrm{sin}}^{2}(\frac{3\pi }{8})$$⇒2sin28π​sin2(82π​)sin2(83π​)

$$\Rightarrow 2{\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{sin}}^{2}(\frac{\pi }{4}){\mathrm{sin}}^{2}(\frac{3\pi }{8})$$⇒2sin28π​sin2(4π​)sin2(83π​)

$$\Rightarrow {\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{sin}}^{2}(\frac{3\pi }{8})$$⇒sin28π​sin2(83π​)

$$\Rightarrow {\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{sin}}^{2}(\frac{\pi }{2}-\frac{\pi }{8})$$⇒sin28π​sin2(2π​−8π​)

$$\Rightarrow {\mathrm{sin}}^{2}\frac{\pi }{8}{\mathrm{cos}}^{2}\frac{\pi }{8}$$⇒sin28π​cos28π​

$$\Rightarrow \frac{1}{4}{(2\mathrm{sin}\frac{\pi }{8}\mathrm{cos}\frac{\pi }{8})}^{2}$$⇒41​(2sin8π​cos8π​)2

$$\frac{1}{4}{\mathrm{sin}}^{2}\frac{\pi }{4}=\frac{1}{8}$$41​sin24π​=81​