Question

Simplify the expression. \(\tan \left(x+\frac{\pi}{2}\right)\)

Short answer

The simplification of \(\tan \left(x+\frac{\pi}{2}\right)\) is \(-\cot(x)\)

Step-by-step solution

Express \(\tan(x)\) in terms of sine and cosine

One can express \(\tan(x)\) as \(\frac{\sin(x)}{\cos(x)}\)

Apply the rule of tangent of sum

The tangent of sum \(\tan(x+y)\) can be expressed as \(\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}\). In this case, y is \(\frac{\pi}{2}\). Now we can rewrite our function as follows \(\tan \left(x+\frac{\pi}{2}\right) = \frac{\tan(x)+\tan(\frac{\pi}{2})}{1-\tan(x).\tan(\frac{\pi}{2})}\).

Simplify trigonometric values

Finally, it's important to remember that \(\tan(\frac{\pi}{2})\) is undefined because \(\cos(\frac{\pi}{2})=0\). Therefore, \(\tan \left(x+\frac{\pi}{2}\right)\) simplifies to \(-\cot(x)\)

Key concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry, often abbreviated as "tan." It is defined as the ratio between the opposite side and the adjacent side in a right-angled triangle. In terms of sine and cosine, it can be expressed as:

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

This ratio is vital when dealing with angles and can be applied in solving various trigonometric problems. The periodic nature of the tangent function means that it repeats its values in cycles, specifically every \(\pi\) radians. Remembering that tangent is undefined whenever

• \( \cos(x) = 0 \)

helps avoid errors, especially during simplification of expressions such as \( \tan \left(x+\frac{\pi}{2}\right) \). That is because \( \tan \left(\frac{\pi}{2}\right) \) becomes undefined. This characteristic of the tangent function is something you’ll encounter often, especially when learning about transformations and sum/difference identities.

Simplifying Expressions

Simplifying expressions in trigonometry is essentially about rewriting them in a more understandable or reduced form. This process involves using trigonometric identities, which are equations involving trigonometric functions that hold true for all values in the function’s domain.
For example, the sum identity for tangent

• \( \tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} \)

allows us to break down complex expressions into more manageable parts. When working with expressions like \( \tan \left(x + \frac{\pi}{2}\right) \), knowing that \( \tan \left(\frac{\pi}{2}\right) \) is undefined is crucial. If you apply this sum formula, it indicates how the expression simplifies despite that undefined aspect. Finding such simplifications helps with solving equations faster and makes the calculations less error-prone.
In this case, learning how to handle undefined values within the expressions leads to insights such as simplifying \( \tan \left(x+\frac{\pi}{2}\right) \) to \(-\cot(x)\), streamlining your work considerably.

Cotangent Function

The cotangent function, abbreviated as "cot," is another critical component of trigonometry. It is essentially the reciprocal of the tangent function. Given the expression:

• \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \)

It represents the ratio of the adjacent side to the opposite side in a right triangle, available as long as \( \sin(x) eq 0 \).
Understanding the relationships between these functions is key when simplifying expressions or solving equations. For instance, when the expression \( \tan \left(x+\frac{\pi}{2}\right) \) was simplified to \(-\cot(x)\), recognizing that \( \cot(x) \) itself is just a different way of expressing tangent's reciprocal highlights the beauty of these relationships.
Utilizing cotangent can help when tangent is undefined, such as with angles like \( \frac{\pi}{2} \). This interplay between the functions allows more versatility in solving problems by converting between forms depending on what is most convenient or simplified for a particular scenario.