To verify the identity [tex]\((\tan(x) + \cot(x))^2 = \sec^2(x) + \csc^2(x)\)[/tex], we will expand and then simplify using trigonometric identities.
1. Expand the left-hand side:
[tex]\[
(\tan(x) + \cot(x))^2 = (\tan(x) + \cot(x)) \cdot (\tan(x) + \cot(x))
\][/tex]
Applying the distributive property (FOIL method):
[tex]\[
(\tan(x) + \cot(x))^2 = \tan^2(x) + \tan(x)\cot(x) + \cot(x)\tan(x) + \cot^2(x)
\][/tex]
Simplifying, knowing that [tex]\(\tan(x)\cot(x) = 1\)[/tex]:
[tex]\[
(\tan(x) + \cot(x))^2 = \tan^2(x) + 2(\tan(x)\cot(x)) + \cot^2(x)
\][/tex]
[tex]\[
(\tan(x) + \cot(x))^2 = \tan^2(x) + 2 \cdot 1 + \cot^2(x)
\][/tex]
[tex]\[
(\tan(x) + \cot(x))^2 = \tan^2(x) + 2 + \cot^2(x)
\][/tex]
2. Right-hand side:
[tex]\[
\sec^2(x) + \csc^2(x)
\][/tex]
Remembering the Pythagorean identities:
[tex]\[
\sec^2(x) = 1 + \tan^2(x) \quad \text{and} \quad \csc^2(x) = 1 + \cot^2(x)
\][/tex]
3. Substitute these identities into the right-hand side:
[tex]\[
\sec^2(x) + \csc^2(x) = (1 + \tan^2(x)) + (1 + \cot^2(x))
\][/tex]
[tex]\[
\sec^2(x) + \csc^2(x) = \tan^2(x) + \cot^2(x) + 1 + 1
\][/tex]
[tex]\[
\sec^2(x) + \csc^2(x) = \tan^2(x) + \cot^2(x) + 2
\][/tex]
Now we compare the simplified left-hand side and right-hand side:
[tex]\[
\tan^2(x) + 2 + \cot^2(x) = \tan^2(x) + \cot^2(x) + 2
\][/tex]
Both sides are equal, hence the identity is verified:
[tex]\[
(\tan(x) + \cot(x))^2 = \sec^2(x) + \csc^2(x)
\][/tex]
