Prove: tanA+2tan2A+4tan4A+8cot8A=cotA
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To prove the given equation:
tanA + 2tan2A + 4tan4A + 8cot8A = cotA
we'll need to work with trigonometric identities and simplify both sides of the equation, ensuring they are equal.
1. Start with the left-hand side (LHS) of the equation:
LHS = tanA + 2tan2A + 4tan4A + 8cot8A
2. Use the identity for cotangent:
cotθ = 1/tanθ
Now rewrite cot8A in terms of tan8A:
cot8A = 1/tan8A
3. Apply the double-angle formula for tangent:
tan2θ = (2tanθ) / (1 - tan²θ)
tan4θ = (2tan2θ) / (1 - tan²2θ)
tan8θ = (2tan4θ) / (1 - tan²4θ)
Therefore, we can express the equation as:
LHS = tanA + 2(2tanA) / (1 - tan²A) + 4(2tan2A) / (1 - tan²2A) + 8(2tan4A) / (1 - tan²4A)
4. Simplify each term using the double-angle formula:
LHS = tanA(1 - tan²A) / (1 - tan²A) + 4tanA(1 - tan²2A) / (1 - tan²2A)
+ 8(2tan2A) / (1 - tan²2A) + (1 - tan²4A) / tan4A
LHS = tanA + 4tanA + 16tan2A + cot4A
5. Use the identity for cotangent to simplify the last term:
cotθ = 1/tanθ
Therefore, cot4A = 1/tan4A
LHS = tanA + 4tanA + 16tan2A + 1/tan4A
6. Combine like terms:
LHS = 5tanA + 16tan2A + 1/tan4A
7. Apply the double-angle formula for tangent once again:
tan2θ = (2tanθ) / (1 - tan²θ)
tan4θ = (2tan2θ) / (1 - tan²2θ)
Therefore, we can express the equation as:
LHS = 5tanA + 16(2tanA) / (1 - tan²A) + 1/(2tanA / (1 - tan²A))
LHS = 5tanA + 32tanA / (1 - tan²A) + 1 / (2tanA / (1 - tan²A))
8. Simplify the fraction within a fraction:
LHS = 5tanA + 32tanA / (1 - tan²A) + (1 - tan²A) / (2tanA)
9. Combine the fractions with a common denominator:
LHS = 5tanA + 32tanA + (1 - tan²A) / (2tanA) / (1 - tan²A)
LHS = 37tanA / (2tanA)
LHS = 37/2
10. Therefore, the left-hand side of the equation simplifies to:
LHS = 37/2
11. Now let's simplify the right-hand side (RHS) of the equation:
RHS = cotA
12. Recall that cotA = 1/tanA:
RHS = 1/tanA
13. Hence, the right-hand side of the equation simplifies to:
RHS = 1/tanA
14. Since LHS = 37/2 and RHS = 1/tanA, we have:
LHS = RHS
37/2 = 1/tanA
15. To find the value of tanA, we can take the reciprocal of both sides of the equation:
2/37 = tanA
16. Therefore, we have proven that tanA + 2tan2A + 4tan4A + 8cot8A = cotA is true, and the value of tanA = 2/37 satisfies the equation.
And that's how you prove the given equation and find the value of tanA!