Question

Prove that tan 78 tan 42 - tan 12 tan 48 = 4

Answer 1

Step-by-step explanation:

We can prove

tanxtan(60∘−x)tan(60∘+x)=tan3x

(Proof below)

Putting x=6∘,

tan6∘tan66∘tan54∘=tan18∘

Putting x=18∘,

tan18∘tan42∘tan78∘=tan54∘

On multiplication,

tan6∘tan66∘tan42∘tan78∘=1

tan6∘tan42∘=cot66∘cot78∘=tan(90∘−66∘)tan(90∘−78∘)=tan24∘tan12∘

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Proof 1:

tanxtan(60∘−x)tan(60∘+x)=tanx⋅tan60∘−tanx1+tan60∘tanx⋅tan60∘+tanx1−tan60∘tanx