Question

Identify ALL equivalent forms of the expression.

tan^2x−10(secx)+26



Select all that apply.


(secx−5)(tanx−5)
(secx−6)(secx−5)
(tanx−5)^2

(secx−5)^2

(secx−5)(secx−5)

Answer 1

To factor the expression \( \tan^2 x - 10 \sec x + 26 \), we first recognize the relationship between tangent and secant. Recall that:

\[
\tan^2 x = \sec^2 x - 1
\]

Substituting this into the expression gives:

\[
\tan^2 x - 10 \sec x + 26 = (\sec^2 x - 1) - 10 \sec x + 26
\]

Simplifying this:

\[
= \sec^2 x - 10 \sec x + 25
\]

We can then rewrite it as:

\[
= \sec^2 x - 10 \sec x + 25 = (\sec x - 5)^2
\]

So we have:

\[
\tan^2 x - 10 \sec x + 26 = (\sec x - 5)^2
\]

Next, let's analyze each option to see which ones are equivalent to our original expression:

1. **\( (secx - 5)(tanx - 5) \)** - This does not match \( (\sec x - 5)^2 \) because it has a factor of \( (tan x - 5) \) which is not a perfect square. **Not equivalent.**

2. **\( (secx - 6)(secx - 5) \)** - This also does not match \( (\sec x - 5)^2 \). Expanding it would lead to different terms. **Not equivalent.**

3. **\( (tanx - 5)^2 \)** - This is also not equivalent since \( \tan^2 x \neq (\sec x - 5)^2 \). **Not equivalent.**

4. **\( (secx - 5)^2 \)** - This matches exactly with our derived expression. **Equivalent.**

5. **\( (secx - 5)(secx - 5) \)** - This is simply another way of writing \( (secx - 5)^2 \). **Equivalent.**

Therefore, the equivalent forms of the expression \( \tan^2 x - 10 \sec x + 26 \) are:

- \( (secx - 5)^2 \)
- \( (secx - 5)(secx - 5) \)

Thus, the final answer is:

- **\( (secx - 5)^2 \)**
- **\( (secx - 5)(secx - 5) \)**