Question

Verify sin^2(x) * cot^4(x) * tan^2(x) = 1 - sin^2(x).

Answer 1

The trigonometric identity sin²(x) × cot´(x) × tan²(x) = 1 - sin²(x) is verified by converting cot(x) and tan(x) to their respective sine and cosine forms and simplifying, ultimately using the Pythagorean identity to confirm the equality.

Step-by-step explanation:

To verify the trigonometric identity sin2(x) × cot4(x) × tan2(x) = 1 - sin2(x), let's convert all the trigonometric functions to sine and cosine.

First, recall the definitions of cotangent and tangent in terms of sine (σ) and cosine (σ) functions:

• cot(x) = cos(x) / sin(x)
• tan(x) = sin(x) / cos(x)

Now let's rewrite the left side of the equation:

1. sin2(x) remains as it is.
2. cot4(x) becomes cos4(x) / sin4(x).
3. tan2(x) becomes sin2(x) / cos2(x).

Combining these, we get:

sin2(x) × (cos4(x) / sin4(x)) × (sin2(x) / cos2(x))

Some terms will cancel out:

sin2(x) × cos4(x) / sin4(x) × sin2(x) / cos2(x) = cos2(x)

Using the Pythagorean identity, which states cos2(x) = 1 - sin2(x), we substitute to get the right side of the original equation.

Therefore, sin2(x) × cot4(x) × tan2(x) = 1 - sin2(x) is verified.