[tex]\tan x(\csc x-\cot x)=[/tex]
(Simplify your answer.)
Answer (2)
Distribute tan x : tan x ( csc x − cot x ) = tan x csc x − tan x cot x .
Rewrite in terms of sin x and cos x : c o s x s i n x ⋅ s i n x 1 − c o s x s i n x ⋅ s i n x c o s x .
Simplify: c o s x 1 − 1 .
Rewrite using sec x : sec x − 1 .
Explanation
Understanding the Problem We are asked to simplify the trigonometric expression tan x ( csc x − cot x ) . To do this, we will rewrite the expression in terms of sin x and cos x and then simplify.
Distributing tan x First, let's distribute tan x to both terms inside the parentheses: tan x ( csc x − cot x ) = tan x csc x − tan x cot x
Rewriting in terms of sin x and cos x 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 , csc x = s i n x 1 , and cot x = s i n x c o s x . Substituting these into the expression, we get: cos x sin x ⋅ sin x 1 − cos x sin x ⋅ sin x cos x
Simplifying the Expression Next, we simplify the expression: cos x sin x ⋅ sin x 1 − cos x sin x ⋅ sin x cos x = cos x 1 − 1
Final Simplification Finally, we rewrite the expression using the definition of sec x = c o s x 1 : cos x 1 − 1 = sec x − 1 Therefore, the simplified expression is sec x − 1 .
Examples
Trigonometric identities are useful in navigation, physics, and engineering. For example, simplifying trigonometric expressions can help determine the angle of elevation needed to launch a projectile to hit a target, or in simplifying calculations involving alternating current in electrical engineering. This problem demonstrates how simplifying trigonometric expressions can lead to more manageable forms for calculations in these fields.
By distributing and rewriting the expression tan x ( csc x − cot x ) in terms of sine and cosine, we simplify it to sec x − 1 . This process utilizes fundamental trigonometric identities and definitions. Therefore, the simplified form of the expression is sec x − 1 .
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