What is the value of $\frac{\cos^2(45^\circ + heta) + \cos^2(45^\circ - heta)}{ an(60^\circ + heta) an(30^\circ - heta)}$

Q: What is the value of $\frac{\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta)}{\tan(60^\circ + \theta)\tan(30^\circ - \theta)}$

What is the value of $\frac{\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta)}{\tan(60^\circ + \theta)\tan(30^\circ - \theta)}$

Question

What is the value of $\frac{\cos^2(45^\circ + 	heta) + \cos^2(45^\circ - 	heta)}{	an(60^\circ + 	heta)	an(30^\circ - 	heta)}$

Accepted Answer

To solve the given expression and verify why Option 3 is correct, we need to evaluate the expression:

$$
\frac{\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta)}{\tan(60^\circ + \theta)\tan(30^\circ - \theta)}
$$

**Step 1: Simplify the numerator**

The numerator is $\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta)$.

Using the identity $\cos^2 A = \frac{1 + \cos 2A}{2}$, we have:

$$
\cos^2(45^\circ + \theta) = \frac{1 + \cos(90^\circ + 2\theta)}{2} = \frac{1 + (-\sin 2\theta)}{2} = \frac{1 - \sin 2\theta}{2}
$$

$$
\cos^2(45^\circ - \theta) = \frac{1 + \cos(90^\circ - 2\theta)}{2} = \frac{1 + \sin 2\theta}{2}
$$

Adding these two expressions:

$$
\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta) = \frac{1 - \sin 2\theta}{2} + \frac{1 + \sin 2\theta}{2} = \frac{2}{2} = 1
$$

**Step 2: Simplify the denominator**

The denominator is $\tan(60^\circ + \theta)\tan(30^\circ - \theta)$.

Using the tangent addition and subtraction formulas:

$$
\tan(60^\circ + \theta) = \frac{\tan 60^\circ + \tan \theta}{1 - \tan 60^\circ \tan \theta} = \frac{\sqrt{3} + \tan \theta}{1 - \sqrt{3} \tan \theta}
$$

$$
\tan(30^\circ - \theta) = \frac{\tan 30^\circ - \tan \theta}{1 + \tan 30^\circ \tan \theta} = \frac{\frac{1}{\sqrt{3}} - \tan \theta}{1 + \frac{1}{\sqrt{3}} \tan \theta}
$$

Multiplying these two expressions:

$$
\tan(60^\circ + \theta)\tan(30^\circ - \theta) = \left(\frac{\sqrt{3} + \tan \theta}{1 - \sqrt{3} \tan \theta}\right) \left(\frac{\frac{1}{\sqrt{3}} - \tan \theta}{1 + \frac{1}{\sqrt{3}} \tan \theta}\right)
$$

This simplifies to:

$$
= \frac{(\sqrt{3} + \tan \theta)(\frac{1}{\sqrt{3}} - \tan \theta)}{(1 - \sqrt{3} \tan \theta)(1 + \frac{1}{\sqrt{3}} \tan \theta)}
$$

The product of the numerators simplifies to 1, and the product of the denominators also simplifies to 1, thus:

$$
\tan(60^\circ + \theta)\tan(30^\circ - \theta) = 1
$$

**Step 3: Evaluate the entire expression**

Now, substituting back into the original expression:

$$
\frac{\cos^2(45^\circ + \theta) + \cos^2(45^\circ - \theta)}{\tan(60^\circ + \theta)\tan(30^\circ - \theta)} = \frac{1}{1} = 1
$$

Therefore, the value of the expression is 1, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (-1):** The expression evaluates to a positive value, not negative, so -1 is incorrect.
- **Option 2 (0):** The expression evaluates to 1, not zero, so 0 is incorrect.
- **Option 4 (2):** The expression evaluates to 1, not 2, so 2 is incorrect.

Thus, the correct answer is indeed Option 3, which is 1.