Value of `cos^(2) 5^(@) + cos^(2) 10^(@) + cos^(2) 80^(@) + cos ^(2) 85^(@) ` is

Options

A. 1
B. 0
C. 2
D. 3

Answer

To find the value of \( \cos^2 5^\circ + \cos^2 10^\circ + \cos^2 80^\circ + \cos^2 85^\circ \), we can use trigonometric identities and properties of angles. Here's a step-by-step solution:

### Step 1: Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \)

We can express \( \cos^2 80^\circ \) and \( \cos^2 85^\circ \) in terms of sine:

\[ \cos^2 80^\circ = 1 - \sin^2 80^\circ \]
\[ \cos^2 85^\circ = 1 - \sin^2 85^\circ \]

### Step 2: Substitute the identities into the expression

Now we can rewrite the original expression:

\[ \cos^2 5^\circ + \cos^2 10^\circ + (1 - \sin^2 80^\circ) + (1 - \sin^2 85^\circ) \]

This simplifies to:

\[ \cos^2 5^\circ + \cos^2 10^\circ + 2 - \sin^2 80^\circ - \sin^2 85^\circ \]

### Step 3: Use the complementary angle identities

Using the identity \( \sin(90^\circ - \theta) = \cos \theta \):

\[ \sin^2 80^\circ = \cos^2 10^\circ \]
\[ \sin^2 85^\circ = \cos^2 5^\circ \]

### Step 4: Substitute these identities back into the expression

Now substitute these back into our expression:

\[ \cos^2 5^\circ + \cos^2 10^\circ + 2 - \cos^2 10^\circ - \cos^2 5^\circ \]

### Step 5: Simplify the expression

Notice that \( \cos^2 5^\circ \) and \( \cos^2 10^\circ \) cancel out:

\[ 2 \]

### Final Answer

Thus, the value of \( \cos^2 5^\circ + \cos^2 10^\circ + \cos^2 80^\circ + \cos^2 85^\circ \) is:

\[ \boxed{2} \]

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