$\tan x(\cot x-\csc x)=$ $\square$ (Simplify your answer.)
Answer (1)
Distribute tan x : tan x cot x − tan x csc x .
Rewrite in terms of sin x and cos x : c o s x s i n x ⋅ s i n x c o s x − c o s x s i n x ⋅ s i n x 1 .
Simplify: 1 − c o s x 1 = 1 − sec x .
The simplified expression is 1 − sec x .
Explanation
Understanding the Problem We are asked to simplify the trigonometric expression tan x ( cot x − csc x ) . To do this, we will use the definitions of the trigonometric functions in terms of sine and cosine.
Distributing tan x First, let's distribute the tan x to both terms inside the parentheses: tan x ( cot x − csc x ) = tan x cot x − tan x csc x
Rewriting in terms of sine and cosine Now, we rewrite the expression in terms of sin x and cos x . Recall that tan x = c o s x s i n x , cot x = s i n x c o s x , and csc x = s i n x 1 . Substituting these into the expression, we get: cos x sin x ⋅ sin x cos x − cos x sin x ⋅ sin x 1
Simplifying the terms Next, we simplify each term. The first term simplifies to: cos x sin x ⋅ sin x cos x = 1 The second term simplifies to: cos x sin x ⋅ sin x 1 = cos x 1 = sec x
Combining the simplified terms Finally, we combine the simplified terms: 1 − sec x Thus, the simplified expression is 1 − sec x .
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and navigation. For example, simplifying trigonometric expressions can help in analyzing wave phenomena, designing electrical circuits, or determining the position and direction of objects in space. In navigation, simplifying expressions involving trigonometric functions can help in calculating distances and angles, which are essential for determining the course of a ship or aircraft.
