Solved Trig Identities worksheet 3.3 name: Prove each | Chegg.com - Free Printable

Here is the step-by-step proof for each of the 8 trigonometric identities on your worksheet. We will use fundamental trigonometric identities:

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Fundamental Identities Used:

- Reciprocal Identities:
- $\csc \theta = \frac{1}{\sin \theta}$
- $\sec \theta = \frac{1}{\cos \theta}$
- $\cot \theta = \frac{\cos \theta}{\sin \theta}$
- $\tan \theta = \frac{\sin \theta}{\cos \theta}$

- Pythagorean Identities:
- $\sin^2 \theta + \cos^2 \theta = 1$
- $1 + \tan^2 \theta = \sec^2 \theta$
- $1 + \cot^2 \theta = \csc^2 \theta$

- Algebraic Manipulations: Factoring, combining fractions, expanding squares.

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1. Prove: $\csc \theta = \frac{\cot \theta}{\cos \theta}$

Proof:

Start with the right-hand side (RHS):

$$
\frac{\cot \theta}{\cos \theta} = \frac{\frac{\cos \theta}{\sin \theta}}{\cos \theta} = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} = \frac{1}{\sin \theta} = \csc \theta
$$

✔ Proved.

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2. Prove: $\frac{1}{\sec^2 x} + \frac{1}{\csc^2 x} = 1$

Proof:

Use reciprocal identities:

$$
\frac{1}{\sec^2 x} = \cos^2 x, \quad \frac{1}{\csc^2 x} = \sin^2 x
$$

So,

$$
\cos^2 x + \sin^2 x = 1 \quad \text{(Pythagorean identity)}
$$

✔ Proved.

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3. Prove: $\csc^2 y \tan^2 y - 1 = \tan^2 y$

Proof:

Start with LHS:

$$
\csc^2 y \tan^2 y - 1
$$

Recall that $\csc^2 y = 1 + \cot^2 y$, but better to write in terms of sin and cos:

$$
\csc^2 y = \frac{1}{\sin^2 y}, \quad \tan^2 y = \frac{\sin^2 y}{\cos^2 y}
$$

So,

$$
\csc^2 y \tan^2 y = \frac{1}{\sin^2 y} \cdot \frac{\sin^2 y}{\cos^2 y} = \frac{1}{\cos^2 y} = \sec^2 y
$$

Thus,

$$
\sec^2 y - 1 = \tan^2 y \quad \text{(Pythagorean identity)}
$$

✔ Proved.

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4. Prove: $\frac{\sec \theta}{\cos \theta} - \frac{\tan \theta}{\cot \theta} = 1$

Proof:

Simplify each term:

First term:
$$
\frac{\sec \theta}{\cos \theta} = \frac{1/\cos \theta}{\cos \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta
$$

Second term:
$$
\frac{\tan \theta}{\cot \theta} = \tan \theta \cdot \tan \theta = \tan^2 \theta \quad \text{(since } \cot \theta = 1/\tan \theta\text{)}
$$

So,

$$
\sec^2 \theta - \tan^2 \theta = 1 \quad \text{(Pythagorean identity)}
$$

✔ Proved.

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5. Prove: $\csc^4 x - \cot^4 x = \csc^2 x + \cot^2 x$

Proof:

Left-hand side is a difference of squares:

$$
\csc^4 x - \cot^4 x = (\csc^2 x)^2 - (\cot^2 x)^2 = (\csc^2 x - \cot^2 x)(\csc^2 x + \cot^2 x)
$$

From Pythagorean identity: $\csc^2 x - \cot^2 x = 1$

So,

$$
1 \cdot (\csc^2 x + \cot^2 x) = \csc^2 x + \cot^2 x
$$

✔ Proved.

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6. Prove: $\sec^4 y - \tan^4 y = \tan^2 y + \sec^2 y$

Proof:

Again, difference of squares:

$$
\sec^4 y - \tan^4 y = (\sec^2 y)^2 - (\tan^2 y)^2 = (\sec^2 y - \tan^2 y)(\sec^2 y + \tan^2 y)
$$

We know: $\sec^2 y - \tan^2 y = 1$

So,

$$
1 \cdot (\sec^2 y + \tan^2 y) = \sec^2 y + \tan^2 y
$$

✔ Proved.

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7. Prove: $(1 - \tan \theta)^2 = \sec^2 \theta - 2\tan \theta$

Proof:

Expand left-hand side:

$$
(1 - \tan \theta)^2 = 1 - 2\tan \theta + \tan^2 \theta
$$

Now, recall: $\sec^2 \theta = 1 + \tan^2 \theta$, so

$$
\sec^2 \theta - 2\tan \theta = (1 + \tan^2 \theta) - 2\tan \theta = 1 - 2\tan \theta + \tan^2 \theta
$$

Which matches the expanded LHS.

✔ Proved.

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8. Prove: $(1 - \sin^2 x)(1 + \tan^2 x) = 1$

Proof:

Use identities:

- $1 - \sin^2 x = \cos^2 x$
- $1 + \tan^2 x = \sec^2 x$

So,

$$
\cos^2 x \cdot \sec^2 x = \cos^2 x \cdot \frac{1}{\cos^2 x} = 1
$$

✔ Proved.

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## ✔ Final Answer Summary:

All 8 identities are proven using basic trigonometric identities and algebraic manipulations.

You can now confidently fill out your worksheet! Let me know if you want to see them written out by hand or explained differently.