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QUESTION:

1. sin⁶theta +cos⁶theta

2. cosec⁶theta - cot⁶theta

3. sec⁸theta -tan⁸theta

4. sin⁴theta- cos⁴theta

ANSWER:

Certainly! I can help you with these trigonometric expressions. Let's go through each one of them step-by-step:

1. sin⁶θ + cos⁶θ:

This expression involves raising both the sine and cosine of theta to the power of 6. However, we can simplify it using a trigonometric identity. We know that sin²θ + cos²θ = 1 (which is the Pythagorean identity). We can square both sides of this equation to get sin⁴θ + 2sin²θcos²θ + cos⁴θ = 1.

Now, let's replace sin⁴θ with (1 - cos⁴θ) from the Pythagorean identity: (1 - cos⁴θ) + 2sin²θcos²θ + cos⁴θ = 1.

Simplifying further, we get 1 + 2sin²θcos²θ = 1. Therefore, sin⁶θ + cos⁶θ = 1 - 2sin²θcos²θ.

2. cosec⁶θ - cot⁶θ:

Here, cosec⁶θ represents (1/sin⁶θ) and cot⁶θ represents (1/tan⁶θ). Let's rewrite the expression: (1/sin⁶θ) - (1/tan⁶θ).

Now, we know that cot⁶θ is equal to (cos⁶θ)/(sin⁶θ). Replacing cot⁶θ with this expression, we get (1/sin⁶θ) - (sin⁶θ)/(cos⁶θ).

To simplify this further, we need to find a common denominator, which is sin⁶θ * cos⁶θ. The expression becomes (cos⁶θ - sin⁶θ)/(sin⁶θ * cos⁶θ).

3. sec⁸θ - tan⁸θ:

Similar to the previous expression, sec⁸θ represents (1/cos⁸θ). We can rewrite the equation: (1/cos⁸θ) - tan⁸θ.

Since tan⁸θ is equal to (sin⁸θ)/(cos⁸θ), we can replace tan⁸θ with this expression: (1/cos⁸θ) - (sin⁸θ)/(cos⁸θ).

To simplify further, we can find a common denominator, which is cos⁸θ. The expression simplifies to (1 - sin⁸θ)/(cos⁸θ).

4. sin⁴θ - cos⁴θ:

This expression involves the difference of two fourth powers. We can use the identity sin²θ - cos²θ = sin(2θ).sin(0) = sin(2θ). Substitute this back into the original equation: (sin²θ - cos²θ)(sin²θ + cos²θ).

Since sin²θ + cos²θ = 1, the expression becomes sin²θ - cos²θ.

I hope this helps!