8 tan square theta - 8 sec square theta

since sec^2 θ - tan^2 θ = 1, you have

-8(sec^2 θ - tan^2 θ) = -8

Better review your basic identities. The most important is, of course,

sin^2 θ + cos^2 θ = 1

divide through by cos^2 and you have

tan^2 θ + 1 = sec^2 θ

Hmm, let me calculate that for you.

8tan²θ - 8sec²θ
= 8(sin²θ / cos²θ) - 8(1 / cos²θ)
= (8sin²θ - 8) / cos²θ

Well, I have to admit, this equation is really trying to tan-gle with our minds!

To simplify the expression "8 tan^2(theta) - 8 sec^2(theta)", we'll start by using trigonometric identities to rewrite tan^2(theta) and sec^2(theta) in terms of sine and cosine.

1. tan^2(theta) = (sin^2(theta)) / (cos^2(theta))
2. sec^2(theta) = 1 / (cos^2(theta))

Now let's substitute these identities into the expression:

8 * [(sin^2(theta)) / (cos^2(theta))] - 8 * [1 / (cos^2(theta))]

Next, we can simplify further by getting a common denominator:

(8 * sin^2(theta) - 8) / cos^2(theta)

Finally, we can factor out a common factor of 8:

8 * (sin^2(theta) - 1) / cos^2(theta)

Alternatively, we can rewrite the expression as:

8 * [sin^2(theta) / cos^2(theta) - 1]

And by using the identity sin^2(theta) + cos^2(theta) = 1, we can simplify it even further:

8 * [1 / cos^2(theta) - 1]

This is the simplified form of the expression "8 tan^2(theta) - 8 sec^2(theta)".

To simplify the given expression, we'll use trigonometric identities.

One identity to remember is:

1 + tan^2(theta) = sec^2(theta).

Notice that the given expression has tan^2(theta) and sec^2(theta). By substituting the above identity, we can rewrite the expression as:

8(1 + tan^2(theta)) - 8 sec^2(theta).

Next, we'll distribute the 8 to both terms:

8 + 8 tan^2(theta) - 8 sec^2(theta).

Now, we have both tan^2(theta) and sec^2(theta) terms.

Another identity to remember is:

tan^2(theta) = sec^2(theta) - 1.

By substituting this identity, we get:

8 + 8 (sec^2(theta) - 1) - 8 sec^2(theta).

Now, we'll simplify further:

8 + 8 sec^2(theta) - 8 - 8 sec^2(theta).

The positive 8 and negative 8 cancel each other out, leaving us with:

0.

Therefore, the simplified expression is 0.