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Chapter 5: Problem 38

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\cos t\left(1+\tan ^{2} t\right)$$

Short Answer

Expert verified
The simplified form of the given expression \( \cos t(1+\tan^2 t) \) is \( 1/\cos t \) or \( \sec t \). Short answer must be a very short answer to the question based on the solution steps

Step by step solution

01

Identify the fundamental identity

The given expression is \( \cos t(1+\tan^2 t) \). We can make this simpler by using the fundamental trigonometric identity \( 1 + \tan^2 t = \sec^2 t \), which states that \( 1 + \tan^2 t \) is equal to \( \sec^2 t \).
02

Substitute the expression

Replace \( 1 + \tan^2 t \) with \( \sec^2 t \) in the given expression. The expression thus becomes \( \cos t * \sec^2 t \).
03

Simplify the expression

We know that \( \sec t \) is equal to \( 1/\cos t \), which implies that \( \sec^2 t = 1/\cos^2 t \). Replace \( \sec^2 t \) with \( 1/\cos^2 t \) in \( \cos t * \sec^2 t \), and we get \( \cos t * (1/\cos^2 t) \). This simplifies to \( 1/\cos t \).

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