`f(x)=sin^2x+cos^4x+2` and `g(x)=cos(cosx)+cos(sinx)` Also let period f(x) and g(x) be `T_1` and `T_2` respectively then

Options

A. `T_(1)=2T_(2)`
B. `2T_(1)=T_(2)`
C. `T_(1)=T_(2)`
D. `T_(1)=4T_(2)`

Explanation

To solve the problem, we need to determine the periods \( T_1 \) and \( T_2 \) of the functions \( f(x) \) and \( g(x) \) respectively.

### Step 1: Analyze the function \( f(x) \)

Given:
\[ f(x) = \sin^2 x + \cos^4 x + 2 \]

We can rewrite \( \cos^4 x \) using the identity \( \cos^2 x = 1 - \sin^2 x \):
\[ \cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2 \]
Expanding this, we get:
\[ \cos^4 x = 1 - 2\sin^2 x + \sin^4 x \]

Substituting this back into \( f(x) \):
\[ f(x) = \sin^2 x + (1 - 2\sin^2 x + \sin^4 x) + 2 \]
\[ f(x) = 3 - \sin^2 x + \sin^4 x \]

### Step 2: Further simplify \( f(x) \)

We can express \( \sin^4 x \) in terms of \( \sin^2 x \):
\[ \sin^4 x = (\sin^2 x)^2 \]

Let \( y = \sin^2 x \):
\[ f(x) = 3 - y + y^2 \]

### Step 3: Find the period \( T_1 \)

The function \( \sin^2 x \) has a period of \( \pi \) and \( \cos^4 x \) can be analyzed as follows:
- The function \( \cos^2 x \) has a period of \( \pi \), thus \( \cos^4 x \) also has a period of \( \pi \).

Since both \( \sin^2 x \) and \( \cos^4 x \) have the same period \( \pi \), the period \( T_1 \) of \( f(x) \) is:
\[ T_1 = \pi \]

### Step 4: Analyze the function \( g(x) \)

Given:
\[ g(x) = \cos(\cos x) + \cos(\sin x) \]

### Step 5: Find the period \( T_2 \)

1. The function \( \cos x \) has a period of \( 2\pi \).
2. The function \( \cos(\cos x) \) will inherit this period since \( \cos x \) is periodic.
3. The function \( \sin x \) also has a period of \( 2\pi \), thus \( \cos(\sin x) \) will also have a period of \( 2\pi \).

Since both components of \( g(x) \) have the same period \( 2\pi \), the period \( T_2 \) of \( g(x) \) is:
\[ T_2 = 2\pi \]

### Step 6: Compare \( T_1 \) and \( T_2 \)

Now we have:
- \( T_1 = \pi \)
- \( T_2 = 2\pi \)

We can see that:
\[ T_1 = \frac{1}{2} T_2 \]
or equivalently,
\[ T_2 = 2 T_1 \]

### Conclusion

Thus, the relationship between the periods is:
\[ T_2 = 2 T_1 \]

### Final Answer

The correct option is that \( T_2 = 2 T_1 \).

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