g) $$ \sec ^{3} x=\frac{\sec ^{3} x-\sec x an ^{2} x}{(1-\operatorname{sen} x)(1+\operatorname{sen} x)} $$

Question

g) $$ \sec ^{3} x=\frac{\sec ^{3} x-\sec x 	an ^{2} x}{(1-\operatorname{sen} x)(1+\operatorname{sen} x)} $$

UpStudy AI · Accepted Answer

To verify the trigonometric identity:

$$
\sec^3 x = \frac{\sec^3 x - \sec x \tan^2 x}{(1 - \sin x)(1 + \sin x)}
$$

let's simplify the right-hand side (RHS) step by step and show that it equals the left-hand side (LHS).

### Step 1: Simplify the Denominator

The denominator of the RHS is:

$$
(1 - \sin x)(1 + \sin x)
$$

This is a difference of squares, which simplifies to:

$$
(1 - \sin x)(1 + \sin x) = 1 - \sin^2 x
$$

Using the Pythagorean identity $$1 - \sin^2 x = \cos^2 x$$, we get:

$$
1 - \sin^2 x = \cos^2 x
$$

### Step 2: Rewrite the RHS with the Simplified Denominator

Substitute the simplified denominator back into the RHS:

$$
\frac{\sec^3 x - \sec x \tan^2 x}{\cos^2 x}
$$

### Step 3: Simplify the Numerator

Recall the definitions of the trigonometric functions:

$$
\sec x = \frac{1}{\cos x}, \quad \tan x = \frac{\sin x}{\cos x}
$$

First, express $$\tan^2 x$$ in terms of $$\sin^2 x$$ and $$\cos^2 x$$:

$$
\tan^2 x = \frac{\sin^2 x}{\cos^2 x}
$$

Now, substitute $$\sec x$$ and $$\tan^2 x$$ into the numerator:

$$
\sec^3 x - \sec x \tan^2 x = \frac{1}{\cos^3 x} - \frac{1}{\cos x} \cdot \frac{\sin^2 x}{\cos^2 x}
$$

Simplify the expression:

$$
\frac{1}{\cos^3 x} - \frac{\sin^2 x}{\cos^3 x} = \frac{1 - \sin^2 x}{\cos^3 x}
$$

Again, using the Pythagorean identity $$1 - \sin^2 x = \cos^2 x$$:

$$
\frac{1 - \sin^2 x}{\cos^3 x} = \frac{\cos^2 x}{\cos^3 x} = \frac{1}{\cos x} = \sec x
$$

### Step 4: Combine the Simplified Numerator and Denominator

Now, the RHS becomes:

$$
\frac{\sec x}{\cos^2 x}
$$

Express $$\sec x$$ in terms of $$\cos x$$:

$$
\frac{\frac{1}{\cos x}}{\cos^2 x} = \frac{1}{\cos^3 x} = \sec^3 x
$$

### Conclusion

After simplifying, both sides of the equation are equal:

$$
\sec^3 x = \sec^3 x
$$

Thus, the given identity is **verified**.
The identity is verified: $$ \sec^3 x = \sec^3 x $$.

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