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Prove that tan 3θ = (3tan θ - tan³θ) / (1 - 3tan²θ).

Aniket Singh

10 Months agoGrade

1 Answer

Askiitians Tutor Team

Approved Tutor Answer10 Months ago

To prove that tan 3θ = (3tan θ - tan³θ) / (1 - 3tan²θ), we can use the tangent addition formula. The formula states that:

Tangent Addition Formula

The tangent of the sum of two angles is given by:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Applying the Formula

Let’s set A = 2θ and B = θ. Then:

tan(3θ) = tan(2θ + θ)

Using the addition formula, we have:

tan(3θ) = (tan 2θ + tan θ) / (1 - tan 2θ tan θ)

Finding tan 2θ

Next, we need to find tan 2θ. Using the double angle formula:

tan 2θ = (2 tan θ) / (1 - tan² θ)

Substituting tan 2θ

Now, substitute tan 2θ into the equation for tan 3θ:

tan(3θ) = ((2 tan θ) / (1 - tan² θ) + tan θ) / (1 - ((2 tan θ) / (1 - tan² θ)) tan θ)

Simplifying the Expression

Combine the terms in the numerator:

tan(3θ) = (2 tan θ + tan θ (1 - tan² θ)) / ((1 - tan² θ) - 2 tan² θ)

This simplifies to:

tan(3θ) = (3 tan θ - tan³ θ) / (1 - 3 tan² θ)

Final Result

Thus, we have shown that:

tan 3θ = (3tan θ - tan³θ) / (1 - 3tan²θ)