Determine the smallest positive value of x(in degrees) for which: tan⁡(x+100∘)=tan⁡(x+50∘)tan⁡(x)tan⁡(x−50∘)

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undodaonePvopxl24 · Accepted Answer

Given tan⁡(x+100∘)=tan⁡(x+50∘)⋅tan⁡(x)⋅tan⁡(x−50∘)⇒(x+100∘)tan⁡(x−50∘)⇒=tan⁡(x+50∘)⋅tan⁡(x∘)⇒sin⁡(x+100∘)⋅cos⁡(x−50∘)cos⁡(x+100∘)⋅sin⁡(x−50∘)Using Componendo and dividendo,=sin⁡(2x+50∘)sin⁡(150∘)=−cos⁡(50∘)cos⁡(2x+50∘)⇒sin⁡(4x+100∘)=−cos⁡(50∘)=−sin⁡(50∘)=sin⁡(180∘+40∘)=sin⁡(360∘−40∘)⇒(4x+100∘)=220∘=320∘So x=30∘ x=55∘

Malia Booth · Accepted Answer

Another way.We need to solvetan⁡(x+100∘)cot⁡x=tan⁡(x−50∘)tan⁡(x+50∘)ortan⁡(x+100∘)cot⁡x−1=tan⁡(x−50∘)tan⁡(x+50∘)−1orsin⁡100∘cos⁡(100∘+x)sin⁡x+cos⁡2xcos⁡(x−50∘)cos⁡(x+50∘)=0orsin⁡100∘(cos⁡2x+cos⁡100∘)+cos⁡2x(sin⁡(2x+100∘)−sin⁡100∘)=0or2sin⁡150∘cos⁡50∘+sin⁡(100∘+4x)=0orsin⁡(100∘+4x)=sin⁡220∘which gives4x=120∘+360∘kwhere k∈Z which isx=30∘+90∘kor4x=−140∘+360∘kwhich isx=−35∘+90∘kand we got the answer: 30∘.

Determine the smallest positive value of x(in degrees)

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