Answered

Verify the identity:

[tex](\tan (x)+\cot (x))^2=\sec ^2(x)+\csc ^2(x)[/tex]

Expand the product, and use the Reciprocal and Pythagorean Identities to simplify:

[tex]
\begin{aligned}
(\tan (x)+\cot (x))^2 &= \tan ^2(x) + 2 \tan (x) \cot (x) + \cot ^2(x) \\
&= \tan ^2(x) + 2 + \cot ^2(x) \\
&= (\tan ^2(x) + 1) + (\cot ^2(x) + 1) \\
&= \sec ^2(x) + \csc ^2(x)
\end{aligned}
\]



Answer :

GinnyAnswer
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]

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