Understanding Trigonometric Identities: Simplifying sin x/cos x cot x
In trigonometry, identities play a crucial role in simplifying and solving various equations. One such identity involves the expression sin x/cos x cot x. This expression can be simplified using fundamental trigonometric relationships. Let's explore this process in detail.
Basics of Trigonometric Identities
Before diving into the simplification, it's important to understand the basic trigonometric identities that are pivotal in solving such expressions:
sin x/cos x tan x cot x 1/tan xSimplifying sin x/cos x cot x
With these identities in mind, let's simplify the expression sin x/cos x cot x. We can follow these steps:
First, recognize that sin x/cos x tan x. The cotangent function is the reciprocal of the tangent function, so cot x 1/tan x. Substitute these identities into the expression sin x/cos x cot x to get: (tan x) * (1/tan x).Now, let's break it down further:
sin x/cos x cot x (tan x) * (1/tan x)
When we multiply tan x and 1/tan x, the tan x in the numerator and denominator cancel each other out, leaving us with:
1
Verification Through Steps
To verify this, we can go through the steps in detail:
Start with sin x/cos x and recognize it simplifies to tan x. Then, since cot x 1/tan x, we substitute it into the expression to get sin x/cos x cot x tan x * 1/tan x. This simplifies to 1.Conclusion
In conclusion, the expression sin x/cos x cot x simplifies to 1 based on the fundamental trigonometric identities and the reciprocal relationship between tangent and cotangent. Understanding these identities can greatly aid in solving more complex trigonometric equations.