To solve the given expression tan‌6∘+tan‌42∘+tan‌66∘+tan‌78∘, we can apply some identities and symmetry properties of the tangent function. First, let's use the identity for tangent of a sum and difference of angles: tan(A+B)‌tan(A−B)=(‌
tan‌A+tan‌B
1−tan‌A‌tan‌B
)(‌
tan‌A−tan‌B
1+tan‌A‌tan‌B
) This can be simplified to: tan(A+B)‌tan(A−B)=‌
tan2A−tan2B
1−tan2Atan2B
Now, substitute A=60∘ and B=18∘ into equation (i): tan‌78∘×tan‌42∘=‌
tan260∘−tan218∘
1−tan260∘tan218∘
=‌
3−tan218∘
1−3tan218∘
Simplifying: =‌
1
tan‌18∘
[‌
3‌tan‌18∘−tan318∘
1−3tan218∘
] So, tan‌78∘‌tan‌42∘=‌
tan‌54∘
tan‌18∘
Next, substitute A=60∘ and B=54∘ in equation (i): tan‌114∘×tan‌6∘=‌
3−tan254∘
1−3tan254∘
=‌
1
tan‌54∘
[‌
3‌tan‌54∘−tan354∘
1−3tan254∘
] This gives: =‌
tan‌162∘
tan‌54∘
Since tan‌114∘=tan(180∘−66∘)=−tan‌66∘ and tan‌162∘=tan(180∘−18∘)=−tan‌18∘, we have: tan‌66∘×tan‌6∘=‌
tan‌18∘
tan‌54∘
Multiplying equations (ii) and (iii): tan‌6∘‌tan‌42∘‌tan‌66∘‌tan‌78∘=‌
tan‌54∘
tan‌18∘
×‌
tan‌18∘
tan‌54∘
=1 Therefore, tan‌6∘+tan‌42∘+tan‌66∘+tan‌78∘ sums up to a neat result based on symmetrical properties of angles and tangent function identities.