1-tan(x+phi/4)/1+tan(x+phi/4) Sama Dengan

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Jun 15, 2024

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Trigonometric Identity: 1-tan(x+φ/4) / 1+tan(x+φ/4) = tan(φ/4 - x)

In this article, we will explore a useful trigonometric identity involving tangent functions.

The Identity

The identity is given by:

$\frac{1-\tan(x+\frac{\phi}{4})}{1+\tan(x+\frac{\phi}{4})} = \tan(\frac{\phi}{4}-x)$

Proof

To prove this identity, we can start by using the tangent sum formula:

$\tan(A+B) = \frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$

Let's set $A = x$ and $B = \frac{\phi}{4}$. Then, we can write:

$\tan(x+\frac{\phi}{4}) = \frac{\tan(x)+\tan(\frac{\phi}{4})}{1-\tan(x)\tan(\frac{\phi}{4})}$

Now, let's multiply both the numerator and the denominator of the right-hand side by $1-\tan(x)\tan(\frac{\phi}{4})$ to get:

$\tan(x+\frac{\phi}{4}) = \frac{\tan(x)(1-\tan(\frac{\phi}{4}))+\tan(\frac{\phi}{4})(1-\tan(x))}{1-\tan^2(x)\tan^2(\frac{\phi}{4})}$

Simplifying the expression, we get:

$\tan(x+\frac{\phi}{4}) = \frac{\tan(x)-\tan^2(x)\tan(\frac{\phi}{4})+\tan(\frac{\phi}{4})-\tan(x)\tan^2(\frac{\phi}{4})}{1-\tan^2(x)\tan^2(\frac{\phi}{4})}$

Now, let's rearrange the terms to get:

$\frac{1-\tan(x+\frac{\phi}{4})}{1+\tan(x+\frac{\phi}{4})} = \frac{1-\tan(x)\tan(\frac{\phi}{4})-\tan(x)-\tan(\frac{\phi}{4})}{1+\tan(x)\tan(\frac{\phi}{4})-\tan(x)+\tan(\frac{\phi}{4})}$

Simplifying further, we get:

$\frac{1-\tan(x+\frac{\phi}{4})}{1+\tan(x+\frac{\phi}{4})} = \frac{\tan(\frac{\phi}{4}-x)}{1+\tan(\frac{\phi}{4}-x)\tan(x)}$

Finally, multiplying both sides by $1+\tan(\frac{\phi}{4}-x)\tan(x)$, we get:

$\frac{1-\tan(x+\frac{\phi}{4})}{1+\tan(x+\frac{\phi}{4})} = \tan(\frac{\phi}{4}-x)$

Thus, we have proved the identity.

Applications

This identity has various applications in trigonometry, calculus, and other areas of mathematics. It can be used to simplify complex trigonometric expressions and to derive other trigonometric identities.

Conclusion

In conclusion, we have proved the trigonometric identity $\frac{1-\tan(x+\frac{\phi}{4})}{1+\tan(x+\frac{\phi}{4})} = \tan(\frac{\phi}{4}-x)$. This identity is a useful tool for simplifying trigonometric expressions and has numerous applications in mathematics.