Math / Trigonometry

Question4) $\cot ^{2} \theta=\frac{\cos ^{2} \theta}{1-\cos ^{2} \theta}$

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.
2. We will use the Pythagorean identity to simplify the expression.
3. We assume $\theta$ is an angle where the trigonometric functions are defined.

STEP 2

1. Recognize and apply trigonometric identities.
2. Simplify the right-hand side of the equation.
3. Verify the equality.

STEP 3

Recognize that the right-hand side of the equation $\frac{\cos^2 \theta}{1 - \cos^2 \theta}$ can be simplified using the Pythagorean identity. The identity states:
$\sin^2 \theta + \cos^2 \theta = 1$
From this, we can express $1 - \cos^2 \theta$ as:
$1 - \cos^2 \theta = \sin^2 \theta$

STEP 4

Substitute the expression for $1 - \cos^2 \theta$ into the right-hand side of the equation:
$\frac{\cos^2 \theta}{1 - \cos^2 \theta} = \frac{\cos^2 \theta}{\sin^2 \theta}$
Recognize that this expression is equivalent to $\cot^2 \theta$ , since:
$\cot \theta = \frac{\cos \theta}{\sin \theta}$
Thus:
$\cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}$

STEP 5

Verify the equality by confirming that both sides of the original equation are indeed equal:
The left-hand side is $\cot^2 \theta$ , and the right-hand side simplifies to $\frac{\cos^2 \theta}{\sin^2 \theta}$ , which is also $\cot^2 \theta$ .
Therefore, the equation holds true:
$\cot^2 \theta = \frac{\cos^2 \theta}{1 - \cos^2 \theta}$
The equality is verified and holds true for the given trigonometric identity.

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