Math / Trigonometry

Question $\frac{1}{1-\sin ^{2} x}=1+\tan ^{2} x$

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.
2. We will use known trigonometric identities to simplify and verify the equation.

STEP 2

1. Simplify the left-hand side using a trigonometric identity.
2. Simplify the right-hand side using a trigonometric identity.
3. Verify that both sides are equal.

STEP 3

Simplify the left-hand side of the equation $\frac{1}{1-\sin^2 x}$ . Recognize that $1 - \sin^2 x$ is a known trigonometric identity:
$1 - \sin^2 x = \cos^2 x$
Thus, the left-hand side becomes:
$\frac{1}{1-\sin^2 x} = \frac{1}{\cos^2 x}$

STEP 4

Simplify the right-hand side of the equation $1 + \tan^2 x$ . Use the trigonometric identity for $\tan^2 x$ :
$\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$
Substitute this into the right-hand side:
$1 + \tan^2 x = 1 + \frac{\sin^2 x}{\cos^2 x}$
Combine the terms over a common denominator:
$1 + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}$
Using the Pythagorean identity $\cos^2 x + \sin^2 x = 1$ , the expression simplifies to:
$\frac{1}{\cos^2 x}$

STEP 5

Verify that both sides of the equation are equal:
The left-hand side is $\frac{1}{\cos^2 x}$ and the right-hand side is also $\frac{1}{\cos^2 x}$ .
Since both sides are equal, the equation is verified to be true.
The equation $\frac{1}{1-\sin^2 x} = 1 + \tan^2 x$ is verified as true.

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