Verifying Trig Identities Worksheet | Math = Love - Free Printable

Let's go through each trigonometric identity one by one, verifying them step-by-step. We'll show that one side of the equation is equivalent to the other using known trigonometric identities and algebraic manipulation.

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1. $-\tan\theta \cos\theta = \sin(-\theta)$

Step 1: Recall that:
- $\tan\theta = \frac{\sin\theta}{\cos\theta}$
- $\sin(-\theta) = -\sin\theta$

Left-hand side (LHS):
$$
-\tan\theta \cos\theta = -\left(\frac{\sin\theta}{\cos\theta}\right)\cos\theta
$$
$$
= -\sin\theta
$$

Right-hand side (RHS):
$$
\sin(-\theta) = -\sin\theta
$$

✔ So, LHS = RHS → Identity verified.

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2. $\cot^2\theta(1 + \tan^2\theta) = \csc^2\theta$

Step 1: Use identity: $1 + \tan^2\theta = \sec^2\theta$

So,
$$
\cot^2\theta(1 + \tan^2\theta) = \cot^2\theta \cdot \sec^2\theta
$$

Now recall:
- $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$
- $\sec\theta = \frac{1}{\cos\theta}$

So:
$$
\cot^2\theta \cdot \sec^2\theta = \left(\frac{\cos^2\theta}{\sin^2\theta}\right)\left(\frac{1}{\cos^2\theta}\right) = \frac{1}{\sin^2\theta} = \csc^2\theta
$$

✔ Verified.

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3. $\frac{\sec\theta}{\csc\theta} = \tan\theta$

Step 1: Write in terms of sine and cosine:
- $\sec\theta = \frac{1}{\cos\theta}$
- $\csc\theta = \frac{1}{\sin\theta}$

So:
$$
\frac{\sec\theta}{\csc\theta} = \frac{1/\cos\theta}{1/\sin\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta
$$

✔ Verified.

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4. $\sin^2\theta(\csc^2\theta - 1) = \cos^2\theta$

Step 1: Recall: $\csc^2\theta = 1 + \cot^2\theta$, so $\csc^2\theta - 1 = \cot^2\theta$

But let's use it directly:
$$
\sin^2\theta(\csc^2\theta - 1) = \sin^2\theta\left(\frac{1}{\sin^2\theta} - 1\right)
$$
$$
= \sin^2\theta \left(\frac{1 - \sin^2\theta}{\sin^2\theta}\right) = 1 - \sin^2\theta = \cos^2\theta
$$

✔ Verified.

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5. $\cot\theta \sin\theta = \cos\theta$

Step 1: $\cot\theta = \frac{\cos\theta}{\sin\theta}$

So:
$$
\cot\theta \sin\theta = \left(\frac{\cos\theta}{\sin\theta}\right)\sin\theta = \cos\theta
$$

✔ Verified.

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

Step 1: Use difference of squares:
$$
(\sec\theta - 1)(\sec\theta + 1) = \sec^2\theta - 1
$$

Recall identity: $\sec^2\theta - 1 = \tan^2\theta$

✔ Verified.

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7. $\sec\theta \cot\theta \sin\theta = 1$

Step 1: Write all in terms of sine and cosine:
- $\sec\theta = \frac{1}{\cos\theta}$
- $\cot\theta = \frac{\cos\theta}{\sin\theta}$

So:
$$
\sec\theta \cot\theta \sin\theta = \left(\frac{1}{\cos\theta}\right)\left(\frac{\cos\theta}{\sin\theta}\right)\sin\theta
$$

Simplify:
- $\frac{1}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} = \frac{1}{\sin\theta}$
- Then multiply by $\sin\theta$: $\frac{1}{\sin\theta} \cdot \sin\theta = 1$

✔ Verified.

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8. $(1 - \cos\theta)(1 + \sec\theta) = \sec\theta - \cos\theta$

Step 1: Expand the left-hand side:

$$
(1 - \cos\theta)(1 + \sec\theta) = 1 \cdot (1 + \sec\theta) - \cos\theta(1 + \sec\theta)
$$
$$
= 1 + \sec\theta - \cos\theta - \cos\theta \cdot \sec\theta
$$

But $\cos\theta \cdot \sec\theta = \cos\theta \cdot \frac{1}{\cos\theta} = 1$

So:
$$
1 + \sec\theta - \cos\theta - 1 = \sec\theta - \cos\theta
$$

✔ Verified.

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9. $\cos\theta \csc\theta = \cot\theta$

Step 1: $\csc\theta = \frac{1}{\sin\theta}$

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

✔ Verified.

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10. $\frac{\cos\theta + \sin\theta}{\sin\theta} = 1 + \cot\theta$

Step 1: Split the fraction:
$$
\frac{\cos\theta + \sin\theta}{\sin\theta} = \frac{\cos\theta}{\sin\theta} + \frac{\sin\theta}{\sin\theta} = \cot\theta + 1
$$

Which is:
$$
1 + \cot\theta
$$

✔ Verified.

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✔ Summary: All 10 identities are verified.

Each identity was proven by applying fundamental trigonometric identities such as:
- $\tan\theta = \frac{\sin\theta}{\cos\theta}$
- $\cot\theta = \frac{\cos\theta}{\sin\theta}$
- $\sec\theta = \frac{1}{\cos\theta}$
- $\csc\theta = \frac{1}{\sin\theta}$
- Pythagorean identities: $\sin^2\theta + \cos^2\theta = 1$, $1 + \tan^2\theta = \sec^2\theta$, etc.

All steps were shown clearly and logically.