If sin 10°sin 50°sin 70°=ktan15°tan45°tan75°, then 8k is equal to

We start by simplifying the given trigonometric expression:

The equation is given as:

\[
\sin 10^\circ \sin 50^\circ \sin 70^\circ = k \tan 15^\circ \tan 45^\circ \tan 75^\circ
\]

### Step 1: Simplify the right-hand side of the equation

\(\tan 45^\circ\) is well known:

\[
\tan 45^\circ = 1
\]

So, the right-hand side simplifies to:

\[
k \times 1 \times \tan 15^\circ \times \tan 75^\circ = k \tan 15^\circ \tan 75^\circ
\]

### Step 2: Simplify the left-hand side of the equation

Next, let's consider the product of sines:

\[
\sin 10^\circ \sin 50^\circ \sin 70^\circ
\]

Using the identity for the product of sine terms at complementary angles:

\[
\sin x \sin (60^\circ - x) \sin (60^\circ + x) = \frac{1}{4} \sin 3x
\]

Here, \(x = 10^\circ\), so:

\[
\sin 10^\circ \sin (60^\circ - 10^\circ) \sin (60^\circ + 10^\circ) = \frac{1}{4} \sin (3 \times 10^\circ) = \frac{1}{4} \sin 30^\circ
\]

We know:

\[
\sin 30^\circ = \frac{1}{2}
\]

So:

\[
\sin 10^\circ \sin 50^\circ \sin 70^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}
\]

### Step 3: Simplify the right-hand side with known trigonometric values

Now, let's consider the right-hand side. We have:

\[
\tan 15^\circ \times \tan 75^\circ
\]

Using the identity:

\[
\tan (90^\circ - x) = \cot x
\]

Thus:

\[
\tan 75^\circ = \cot 15^\circ
\]

So the product becomes:

\[
\tan 15^\circ \times \tan 75^\circ = \tan 15^\circ \times \cot 15^\circ = 1
\]

This simplifies the right-hand side to \(k\).

### Step 4: Equate both sides and solve for \(k\)

Given the equation:

\[
\frac{1}{8} = k
\]

We need to find \(8k\):

\[
8k = 8 \times \frac{1}{8} = 1
\]

Thus, \(8k\) is equal to \(1\).

### Final Answer:

\[
\boxed{1}
\]

Would you like to delve into any part of this solution? Here are 5 related questions for further exploration:

1. How can we generalize the product of sine functions for other angles?
2. What are the other trigonometric identities that involve complementary angles?
3. How does the identity \(\tan(90^\circ - x) = \cot x\) help in simplifying trigonometric products?
4. What is the significance of the sine and tangent values of specific angles in trigonometry?
5. Can we prove the sine product identity used in this problem geometrically?

**Tip:** Always remember to simplify trigonometric expressions using known identities first before diving into complex calculations. This often leads to quicker and more elegant solutions.

Explore how to solve the trigonometric equation sin 10°sin 50°sin 70° = k tan 15°tan 45°tan 75° and find the value of 8k. This problem involves applying trigonometric identities and simplifying expressions. Suitable for high school students studying trigonometry.

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