Tan2x Identity

Trigonometry, as it suggests, is all about triangles, mostly right-angled triangles. Trigonometry is a system that helps us to work out missing or unknown side lengths or angles in a triangle.

Sine, Cosine & Tangent are some of the functions used for finding the sides of a triangle with their given universal values. In this article, we are going to look at tan2x identity which is yet another important identity in trigonometry.

History

The Greek mathematician Hipparchus produced the first known table of chords in about 140 BC. Although these tables have not survived, it is claimed that Hipparchus wrote twelve books of tables of chords, making Hipparchus the founder of trigonometry.

Tan2x Formula 

Tan2x identity is one of the important identities in trigonometry that can be expressed in different ways. 

Tangent(2x) can be expressed in terms of different trigonometric functions such as sine, cosine and tangent.

\(\tan(2x) = \frac{\sin2x}{\cos2x}\)

\(\tan(2x) = \frac{2\tan2x}{1 – \tan^2x}\)

Derivations

1. \(\tan(2x) = \frac{\sin2x}{\cos2x}\)

{as we know that \(\tan(x) = \frac{sin(x)}{cos(x)}\) }

\(\tan(2x) = \frac{\sin(2x)}{\cos(2x)}\)

\(\tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2x – \sin^2x}\)

2. \(\tan(2x)     = \frac{2\tan(x)}{1 – \tan^2x}\)

\(\tan(2x) = \tan(x + x)\)

\(\tan(2x) = \frac{2 \tan(x)}{1 – \tan^2x}\)

Questions

Question 1. Determine the value of \(\tan2x\) if \(\tan x = \frac{1}{2}\).

Solution. We have, \(\tan(x) = \frac{1}{2}\)

\(\tan^2(x) = \frac{1}{4}\)

Therefore, \(\tan(2x) = \frac{2 \tan(x)}{1 – \tan^2x}\)

\(\tan2x = \frac{2(\frac{1}{2})}{1 – \frac{1}{4}} = \frac{4}{3}\)

Question 2. Find the value of \(\tan2x\) if \(\sin x = \frac{1}{2}\) and \(\cos x = \frac{\sqrt{3}}{2}\)

Solution. \(\tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2x – \sin^2x}\)

\(\tan2x  = \frac{2(\frac{1}{2})(\frac{\sqrt{3}}{2})}{(\frac{\sqrt{3}}{2})^2 – (\frac{1}{2})^2)}\)

\(\tan2x = \sqrt{3}\)

FAQs

What are the formulas for tan2x identity?

\(\tan(2x) = \frac{\sin2x}{\cos2x}\)
\(\tan(2x)= \frac{2\tan2x}{1 – \tan^2x}\)

What is the derivative of tan2x?

Derivative of \(\tan2x = 2sec^2(2x)\)

Tan2x formula in tan(x) terms.

\(\tan(2x) = \frac{2 \tan(x)}{1 – \tan^2x}\)

What is the integral of tan(2x)?

The integral of \(\tan(2x) = \frac{1}{2} \ln|\sec 2x| + C\).

Related Articles

  1. Sin2x Identity
  2. Cos2x Identity
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