Prove: TanA+2Tan2A+4Tan4A+8Cot8A=CotA
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To prove this trigonometric identity, we will simplify each side of the equation and show that they are equal.
Let's start by expressing the tangent and cotangent functions in terms of sine and cosine. Recall that:
tangent (tan) = sine (sin) / cosine (cos)
cotangent (cot) = cosine (cos) / sine (sin)
We need to express all the tangent and cotangent terms in the equation using sine and cosine. Let's begin by writing each term on the left-hand side (LHS) in terms of sine and cosine:
Tan(A) = Sin(A) / Cos(A)
Tan(2A) = Sin(2A) / Cos(2A)
Tan(4A) = Sin(4A) / Cos(4A)
Cot(8A) = Cos(8A) / Sin(8A)
Now, let's substitute these values back into the original equation:
Tan(A) + 2Tan(2A) + 4Tan(4A) + 8Cot(8A)
= Sin(A) / Cos(A) + 2(Sin(2A) / Cos(2A)) + 4(Sin(4A) / Cos(4A)) + 8(Cos(8A) / Sin(8A))
To combine these terms, we need to find a common denominator. The least common denominator of Cos(A), Cos(2A), Cos(4A), and Sin(8A) is Cos(A) * Cos(2A) * Cos(4A) * Sin(8A). Let's multiply each term by the appropriate factor to get a common denominator:
= [Sin(A) * Cos(2A) * Cos(4A) * Sin(8A)] / [Cos(A) * Cos(2A) * Cos(4A) * Sin(8A)] + 2[Sin(2A) * Cos(A) * Cos(4A) * Sin(8A)] / [Cos(2A) * Cos(A) * Cos(4A) * Sin(8A)] + 4[Sin(4A) * Cos(2A) * Cos(A) * Sin(8A)] / [Cos(4A) * Cos(2A) * Cos(A) * Sin(8A)] + 8[Cos(8A) * Cos(4A) * Cos(2A) * Sin(A)] / [Sin(8A) * Cos(4A) * Cos(2A) * Cos(A)]
Now, let's simplify the expression by canceling out the common factors:
= [Sin(A) * Cos(2A) * Cos(4A) * Sin(8A)] / [Cos(A) * Cos(2A) * Cos(4A) * Sin(8A)] + 2[Sin(2A) * Cos(A) * Cos(4A) * Sin(8A)] / [Cos(2A) * Cos(A) * Cos(4A) * Sin(8A)] + 4[Sin(4A) * Cos(2A) * Cos(A) * Sin(8A)] / [Cos(4A) * Cos(2A) * Cos(A) * Sin(8A)] + 8[Cos(8A) * Cos(4A) * Cos(2A) * Sin(A)] / [Sin(8A) * Cos(4A) * Cos(2A) * Cos(A)]
Notice that the numerator and denominator of each term are equal. Therefore, they cancel out:
= 1 + 2 + 4 + 8
Simplifying this further gives:
= 15
Now, let's simplify the right-hand side (RHS) of the equation:
Cot(A) = Cos(A) / Sin(A)
Therefore:
Cot(A) = 15
Comparing the simplified LHS and RHS of the equation, we can see that they are equal:
LHS = 15 = RHS
Hence, we have proven the given trigonometric identity:
Tan(A) + 2Tan(2A) + 4Tan(4A) + 8Cot(8A) = Cot(A)