The number of elements in the set { x [ 0 , 180 ° ] : tan ( x + 100 ° ) = tan ( x + 50 ° ) tan ( x - 50 ° ) } is __________ . [2026]

Question

The number of elements in the set ${x \in [0, 180 °] : \tan (x + 100 °) = \tan (x + 50 °) \tan x \tan (x - 50 °)}$ is __________ . [2026]

Smart Achievers · Accepted Answer

(4)

$\frac{\tan (x + 100 °)}{\tan x} = \tan (x + 50 °) \tan (x - 50 °)$

$\frac{\sin (x + 100 °) \cos x}{\cos (x + 100 °) \sin x} = \frac{\sin (x + 50 °) \sin (x - 50 °)}{\cos (x + 50 °) \cos (x - 50 °)}$

$Apply C & D$

$\frac{\sin (2 x + 100 °)}{\sin 100 °} = \frac{\cos 100 °}{- \cos 2 x}$

$2 \sin (2 x + 100 °) \cos 2 x + \sin 200 ° = 0$

$\sin (4 x + 100 °) + \sin 100 ° + \sin 200 ° = 0$

$\sin (4 x + 100 °) = - 2 \sin 150 ° \cos 50 °$

$\sin (4 x + 100 °) = - \cos 50 ° = \sin (- 40 °)$

$∴$ $4 x + 100 ° = n π + {(- 1)}^{n} \cdot (- 40 °)$

$x = \frac{n π + {(- 1)}^{n + 1} (40 °) - 100 °}{4}$

$∴$ $x = 30 °, 55 °, 120 °, 145 ° in (0, π)$

$∴$ $no. of solutions = 4$

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