Simplify the expression [tex]\sqrt{25 x^2 - 100}[/tex] as much as possible after substituting [tex]x = 2 \sec(\theta)[/tex]. Assume [tex]0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}[/tex].

Hint: Use "theta" for [tex]\theta[/tex].



Answer :

Let's begin by substituting [tex]\( x = 2 \sec(\theta) \)[/tex] into the expression [tex]\( \sqrt{25 x^2 - 100} \)[/tex].

First, substitute [tex]\( x = 2 \sec(\theta) \)[/tex] into the expression:

[tex]\[ 25 x^2 - 100 \][/tex]

This becomes:

[tex]\[ 25 (2 \sec(\theta))^2 - 100 \][/tex]

Next, calculate [tex]\( (2 \sec(\theta))^2 \)[/tex]:

[tex]\[ (2 \sec(\theta))^2 = 4 \sec^2(\theta) \][/tex]

So, the expression transforms into:

[tex]\[ 25 \cdot 4 \sec^2(\theta) - 100 \][/tex]

Simplify the terms inside the parenthesis:

[tex]\[ 100 \sec^2(\theta) - 100 \][/tex]

Factor out the common factor of 100:

[tex]\[ 100 (\sec^2(\theta) - 1) \][/tex]

Now, recall the Pythagorean identity for secant:

[tex]\[ \sec^2(\theta) = 1 + \tan^2(\theta) \][/tex]

So:

[tex]\[ \sec^2(\theta) - 1 = \tan^2(\theta) \][/tex]

Thus, the expression simplifies to:

[tex]\[ 100 (\tan^2(\theta)) \][/tex]

Now, take the square root of the entire expression:

[tex]\[ \sqrt{100 \tan^2(\theta)} \][/tex]

This can be further simplified by taking the square root of each term:

[tex]\[ \sqrt{100} \cdot \sqrt{\tan^2(\theta)} \][/tex]

[tex]\[ 10 \cdot |\tan(\theta)| \][/tex]

Since [tex]\(\tan(\theta)\)[/tex] is positive in the interval [tex]\( 0^\circ < \theta < 90^\circ \)[/tex], we can drop the absolute value:

[tex]\[ 10 \tan(\theta) \][/tex]

Thus, the simplified expression is:

[tex]\[ \boxed{10 \tan(\theta)} \][/tex]