$\tan 12°+\tan 78°=$

This trigonometric expression involves simplifying the sum of tangent functions at specific angles, 12° and 78°. Using trigonometric identities, we can simplify the expression step by step.

Step 1: Using the complementary angle identity

We know that: tan(90° - θ) = cot θ
So, tan 78° = cot 12°.

Now substitute: 
tan 12° + tan 78° = tan 12° + cot 12°

Step 2: Express cotangent in terms of sine and cosine

cot 12° = cos 12° / sin 12°
Substitute into the equation: 
tan 12° + cot 12° = (sin 12° / cos 12°) + (cos 12° / sin 12°)

Step 3: Combine terms using a common denominator

tan 12° + cot 12° = (sin² 12° + cos² 12°) / (sin 12° cos 12°)

Step 4: Simplify using the Pythagorean identity

The identity sin² θ + cos² θ = 1 gives: 
tan 12° + cot 12° = 1 / (sin 12° cos 12°)

Step 5: Express in terms of double-angle identity

Using the double-angle identity: 
sin 2θ = 2 sin θ cos θ
Rewriting: 
sin 24° = 2 sin 12° cos 12°

So: 
1 / (sin 12° cos 12°) = 2 / sin 24°

Final Answer

tan 12° + tan 78° = 2 csc 24°

Summary

The simplification of tan 12° + tan 78° involves recognizing complementary angles, using trigonometric identities, and expressing the result in terms of the cosecant function. The final result is 2 csc 24°.