Prove the rollowing Trig identtres
a) 1+
tan- 0
sin? 0
b) sin' 0 + cos? 0 + tan? 0 = cos? e

Let's correct and then prove the given trigonometric identities.

### a) \(\sin^2 \theta + \cos^2 \theta = 1\)

This is a fundamental trigonometric identity.

**Proof:**

The Pythagorean Theorem relates the sides of a right triangle, which can be expressed using trigonometric functions for an angle \(\theta\):

\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]

According to the Pythagorean Theorem:

\[ (\text{opposite})^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2 \]

Dividing each term by \((\text{hypotenuse})^2\), we get:

\[ \left(\frac{\text{opposite}}{\text{hypotenuse}}\right)^2 + \left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)^2 = 1 \]

Which simplifies to:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

This completes the proof for the first identity.

### b) \(\sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sec^2 \theta\)

We start with the known identities:

1. \(\sin^2 \theta + \cos^2 \theta = 1\)
2. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
3. \(\sec \theta = \frac{1}{\cos \theta}\)

Using these, we can express \(\tan^2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\):

\[ \tan^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta} \]

Now let's rewrite the left-hand side of the given identity:

\[ \sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sin^2 \theta + \cos^2 \theta + \frac{\sin^2 \theta}{\cos^2 \theta} \]

We already know that:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

So, the equation becomes:

\[ 1 + \frac{\sin^2 \theta}{\cos^2 \theta} \]

We also know that:

\[ \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta \]

Therefore, we have:

\[ 1 + \tan^2 \theta \]

Using the identity:

\[ 1 + \tan^2 \theta = \sec^2 \theta \]

So,

\[ \sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sec^2 \theta \]

This completes the proof for the second identity.

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Would you like more details on these proofs or have any questions? Here are some other questions you might consider:

1. How do you derive the double angle formulas in trigonometry?
2. Can you explain the Law of Sines and its applications?
3. How do you solve a trigonometric equation involving multiple angles?
4. What is the proof of the addition formulas for sine and cosine?
5. How do you find the general solution of a trigonometric equation?

**Tip:** When solving trigonometric identities, always start by writing known identities and expressing everything in terms of sine and cosine for simplification.

Explore how to prove trigonometric identities involving sine, cosine, and tangent. Includes detailed steps and explanations for proving fundamental identities like sin^2 theta + cos^2 theta = 1 and more complex identities. Suitable for high school students studying trigonometry.

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