The value of 2sin2 π/6 + cosec2 7π/6 x cos2 π/3 is m/m - 1. Find

1.	The value of 2sin2 π/6 + cosec2 7π/6 x cos2 π/3 is m/m - 1. Find the value of 'm'.
Answer» 2sin² \(\cfrac{\pi}6\) + cosec² \(\cfrac{7\pi}6\) cos² \(\cfrac{\pi}3\) = 2(sin \(\cfrac{\pi}6\))² + (cosec (π + \(\cfrac{\pi}6\)))² (cos\(\cfrac{\pi}3\))² = 2 (\(\cfrac{1}2\))²+ (-cosec\(\cfrac{\pi}6\))²(\(\cfrac{1}2\))² (\(\because\) sin \(\cfrac{\pi}6\) = cos \(\cfrac{\pi}3\) = \(\cfrac{1}2\)) = \(\cfrac{2}4\) + (-2)² x \(\cfrac{1}4\) (\(\because\) cosec \(\cfrac{\pi}6\) = \(\cfrac1{sin{\cfrac{\pi}6}}\) = \(\cfrac1{{\cfrac{1}2}}\) = 2) = \(\cfrac{1}2\) + 4 x \(\cfrac{1}4\) = \(\cfrac{1}2\) + 1 = \(\cfrac{3}2\) = \(\cfrac{3}{3-1}\) ∴ m = 3 (by comparing)

The value of 2sin2 π/6 + cosec2 7π/6 x cos2 π/3 is m/m - 1. Find the value of 'm'.

Answer»

2sin² \(\cfrac{\pi}6\) + cosec² \(\cfrac{7\pi}6\) cos² \(\cfrac{\pi}3\)

= 2(sin \(\cfrac{\pi}6\))² + (cosec (π + \(\cfrac{\pi}6\)))² (cos\(\cfrac{\pi}3\))²

= 2 (\(\cfrac{1}2\))²+ (-cosec\(\cfrac{\pi}6\))²(\(\cfrac{1}2\))² (\(\because\) sin \(\cfrac{\pi}6\) = cos \(\cfrac{\pi}3\) = \(\cfrac{1}2\))

= \(\cfrac{2}4\) + (-2)² x \(\cfrac{1}4\) (\(\because\) cosec \(\cfrac{\pi}6\) = \(\cfrac1{sin{\cfrac{\pi}6}}\) = \(\cfrac1{{\cfrac{1}2}}\) = 2)

= \(\cfrac{1}2\) + 4 x \(\cfrac{1}4\) = \(\cfrac{1}2\) + 1

= \(\cfrac{3}2\) = \(\cfrac{3}{3-1}\)

∴ m = 3 (by comparing)

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