To find the value of the expression $\cos 10^\circ \times \cos 30^\circ \times \cos 50^\circ \times \cos 70^\circ$, we can use known trigonometric values and identities. This problem involves evaluating a product of cosine functions.

Finding the Cosine Product Value

Let the given expression be represented by $P$.

$ P = \cos 10^\circ \times \cos 30^\circ \times \cos 50^\circ \times \cos 70^\circ $

We know the specific value for $\cos 30^\circ$. Substituting this known value simplifies the expression:

$ \cos 30^\circ = \frac{\sqrt{3}}{2} $

Replacing $\cos 30^\circ$ in the expression for $P$, we get:

$ P = \frac{\sqrt{3}}{2} \times (\cos 10^\circ \times \cos 50^\circ \times \cos 70^\circ) $

Applying Trigonometric Identities for Cosine Product

To evaluate the remaining product, $\cos 10^\circ \times \cos 50^\circ \times \cos 70^\circ$, we can rearrange the terms and apply a useful trigonometric identity. Notice that $50^\circ$ can be written as $60^\circ - 10^\circ$ and $70^\circ$ can be written as $60^\circ + 10^\circ$.

• $\cos 50^\circ = \cos(60^\circ - 10^\circ)$
• $\cos 70^\circ = \cos(60^\circ + 10^\circ)$

The product $\cos 10^\circ \times \cos 50^\circ \times \cos 70^\circ$ can be rewritten as:

$ \cos 10^\circ \times \cos(60^\circ - 10^\circ) \times \cos(60^\circ + 10^\circ) $

This matches the form of the following trigonometric identity:

$ \cos \theta \times \cos(60^\circ - \theta) \times \cos(60^\circ + \theta) = \frac{1}{4} \cos(3\theta) $

By setting $\theta = 10^\circ$, we can apply this identity directly to our expression:

$ \cos 10^\circ \times \cos 50^\circ \times \cos 70^\circ = \frac{1}{4} \cos(3 \times 10^\circ) $

Calculating the argument of the cosine:

$ = \frac{1}{4} \cos(30^\circ) $

Using the known value $\cos 30^\circ = \frac{\sqrt{3}}{2}$, we get:

$ = \frac{1}{4} \times \frac{\sqrt{3}}{2} $

$ = \frac{\sqrt{3}}{8} $

Step-by-Step Calculation of the Full Expression

Now, we substitute the calculated value of $\cos 10^\circ \times \cos 50^\circ \times \cos 70^\circ$ back into our expression for $P$:

$ P = \frac{\sqrt{3}}{2} \times \left( \frac{\sqrt{3}}{8} \right) $

Performing the multiplication of the two fractions:

$ P = \frac{\sqrt{3} \times \sqrt{3}}{2 \times 8} $

Simplifying the numerator and the denominator:

$ P = \frac{3}{16} $

Final Result for the Cosine Product

Therefore, the value of the given trigonometric expression $\cos 10^\circ \times \cos 30^\circ \times \cos 50^\circ \times \cos 70^\circ$ is $\frac{3}{16}$.