The value of

$$
\left(\frac{\sin heta - 2\sin^3 heta}{2\cos^3 heta - \cos heta} ight)^3 \cdot \frac{1}{ an heta} - \sec^2 heta
$$

is:

Question

The value of

$$
\left(\frac{\sin	heta - 2\sin^3	heta}{2\cos^3	heta - \cos	heta}ight)^3 \cdot \frac{1}{	an	heta} - \sec^2	heta
$$

is:

Accepted Answer

We are given the expression:

$$
\left( \frac{\sin	heta - 2\sin^3	heta}{2\cos^3	heta - \cos	heta} ight)^3 \cdot \frac{1}{	an	heta} - \sec^2	heta
$$

---

### Step 1: Simplify the numerator and denominator inside the brackets

#### **Numerator:**

$$
\sin	heta - 2\sin^3	heta = \sin	heta(1 - 2\sin^2	heta)
$$

But recall:

$$
\sin^2	heta = 1 - \cos^2	heta
\Rightarrow 1 - 2\sin^2	heta = 1 - 2(1 - \cos^2	heta) = 2\cos^2	heta - 1
$$

So the numerator becomes:

$$
\sin	heta(2\cos^2	heta - 1)
$$

---

#### **Denominator:**

$$
2\cos^3	heta - \cos	heta = \cos	heta(2\cos^2	heta - 1)
$$

---

### Step 2: Substitute back into the expression

So the entire expression becomes:

$$
\left( \frac{\sin	heta(2\cos^2	heta - 1)}{\cos	heta(2\cos^2	heta - 1)} ight)^3 \cdot \frac{1}{	an	heta} - \sec^2	heta
$$

Cancel $2\cos^2	heta - 1$ (assuming it's non-zero):

$$
\left( \frac{\sin	heta}{\cos	heta} ight)^3 \cdot \frac{1}{	an	heta} - \sec^2	heta
\Rightarrow (	an	heta)^3 \cdot \frac{1}{	an	heta} - \sec^2	heta
= 	an^2	heta - \sec^2	heta
$$

---

### Step 3: Use identity

We know:

$$
	an^2	heta = \sec^2	heta - 1
\Rightarrow 	an^2	heta - \sec^2	heta = (\sec^2	heta - 1) - \sec^2	heta = -1
$$

---

### ✅ Final Answer:

$$
\boxed{-1}
$$

Answer

Answer

Answer