Pregunta
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The expression simplifies to
.
Solución
Let’s simplify the given trigonometric expression step by step:
Step 1: Simplify Each Trigonometric Function
-
Simplify
: The tangent function has a period of, meaning: -
Simplify
: The cosine function has a period of, so: (Recall thatbecause cosine is an even function.) -
Simplify
: Break downinto : Using the cosine addition formula:
Step 2: Substitute Simplified Functions Back into the Expression
Step 3: Simplify the Fraction
-
Express
in terms of and : -
Substitute
back into the expression: -
Simplify the fraction:
-
Add the constant term:
Final Answer
The expression simplifies to:
Note: This simplification holds for all values of
where
, as division by zero is undefined.
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Como
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Beyond the Answer
First, let’s simplify the expression step by step.
-
Simplifying
: because tangent has a period of. -
Simplifying
: since cosine is an even function andis effectively to rotations. -
Simplifying
: becauseis beyond a full rotation, and is the angle we need to convert.
Now substituting these simplifications back into the original expression:
This can be further simplified:
Using
, we can substitute
:
This simplifies to:
For
:
Thus, if
, the final result is:

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