The number of elements in the set (x ∈ (0 ,180^ ) : tan(x+100^ )= tan(x+50^ ) tanxtan(x-50^ )) is __________.

Question

Accepted Answer

Correct answer: 4. (4) tan(x+100^ ) tanx= tan(x+50^ ) tan(x-50^ ) sin(x+100^ ) cosx cos(x+100^ ) sinx= sin(x+50^ ) sin(x-50^ ) cos(x+50^ ) cos(x-50^ )Applying Componendo & Dividendo: sin(2x+100^ ) sin100^ = cos100^ - cos2x 2 sin(2x+100^ ) cos2x+ sin200^ =0 sin(4x+100^ )+ sin100^ + sin200^ =0 sin(4x+100^ )=-2 sin ⁡150^ cos50^ sin(4x+100^ )=- cos50^ = sin(-40^ ) ∴4x+100^ = nπ+ (-1 )^n(-40^ ) x= nπ+ (-1 )^n+1 (40^ )-100^ 4 ∴x=30^ ,55^ ,120^ ,145^ in ( 0 , ) ∴ Number of solutions =4

$The number of elements in the set (x \in (0, 180^{\circ}) : tan (x + 100^{\circ}) = tan (x + 50^{\circ}) tanxtan (x - 50^{\circ})) is __________.$

Solution

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