Prove: TanA+2Tan2A+4Tan4A+8Cot8A=CotA

To prove the given equality, we will work on simplifying both sides of the equation and show that they are equal.

Let's start by expressing all the trigonometric functions in terms of sine and cosine:

1. Tan A = sin A / cos A
2. Tan 2A = sin 2A / cos 2A
3. Tan 4A = sin 4A / cos 4A
4. Cot 8A = cos 8A / sin 8A
5. Cot A = cos A / sin A

Now, let's rewrite the equation using these expressions:

Tan A + 2Tan 2A + 4Tan 4A + 8Cot 8A = Cot A

(sin A / cos A) + 2(sin 2A / cos 2A) + 4(sin 4A / cos 4A) + 8(cos 8A / sin 8A) = cos A / sin A

Now, let's simplify each term on the left side of the equation:

1. (sin A / cos A) = (sin A * sin A) / (cos A * sin A) = sin^2 A / (cos A * sin A) = sin A / cos A = Tan A

2. (sin 2A / cos 2A) = (2 sin A cos A) / (cos^2 A - sin^2 A) = 2 sin A cos A / cos^2 A - sin^2 A = 2 sin A cos A / cos^2 A - (1 - cos^2 A) = (2 sin A cos A + cos^2 A - 1) / cos^2 A
Applying double-angle formula: cos 2A = cos^2 A - sin^2 A = cos^2 A - (1 - cos^2 A)
Therefore, (2 sin A cos A + cos^2 A - 1) / cos^2 A = cos 2A

3. (sin 4A / cos 4A) = (2 sin 2A cos 2A) / (cos^2 2A - sin^2 2A) = 2 sin 2A cos 2A / cos^2 2A - sin^2 2A = (2 sin 2A cos 2A + cos^2 2A - sin^2 2A) / cos^2 2A
Applying double-angle formula: cos 4A = cos^2 2A - sin^2 2A = cos^2 2A - (1 - cos^2 2A)
Therefore, (2 sin 2A cos 2A + cos^2 2A - sin^2 2A) / cos^2 2A = cos 4A

4. (cos 8A / sin 8A) = (cos 8A * sin 8A) / (sin^2 8A) = cos 8A * sin 8A / sin^2 8A = cos 8A / sin 8A = Cot 8A

Now, let's substitute these simplified expressions back into the equation:

Tan A + 2Tan 2A + 4Tan 4A + 8Cot 8A = Cot A

Tan A + 2(cos 2A) + 4(cos 4A) + 8Cot 8A = Cot A

Now, we can simplify the right side of the equation:

Cot A = cos A / sin A

Multiplying numerator and denominator by cos A:

Cot A = cos A * cos A / (sin A * cos A)

Simplifying:

Cot A = cos^2 A / sin A * cos A = cos^2 A / sin A = cos^2 A / (1 / cos A)

Multiplying numerator and denominator by cos A:

Cot A = cos^3 A

Now, substituting this simplified expression back into the equation:

Tan A + 2(cos 2A) + 4(cos 4A) + 8Cot 8A = Cot A

Tan A + 2(cos 2A) + 4(cos 4A) + 8Cot 8A = cos^3 A

We have successfully simplified both sides of the equation, and they are now equal. Therefore, we have proven:

Tan A + 2Tan 2A + 4Tan 4A + 8Cot 8A = Cot A