The equation given:

$\sin 20 \cdot \tan 20 + \sin 20 \cdot \tan 20 = 1$

can be simplified and analyzed step-by-step:

Step 1: Combine like terms

Since $\sin 20 \cdot \tan 20$ appears twice, we can write: $2 \cdot \sin 20 \cdot \tan 20 = 1$

Step 2: Solve for $\sin 20 \cdot \tan 20$

Divide both sides by 2: $\sin 20 \cdot \tan 20 = \frac{1}{2}$

Step 3: Relationship between $\tan 20$ and $\sin 20$

We know: $\tan x = \frac{\sin x}{\cos x}$ So, substituting $\tan 20 = \frac{\sin 20}{\cos 20}$: $\sin 20 \cdot \tan 20 = \sin 20 \cdot \frac{\sin 20}{\cos 20}$ $\sin 20 \cdot \tan 20 = \frac{\sin^2 20}{\cos 20}$

Step 4: Substitute back

We now have: $\frac{\sin^2 20}{\cos 20} = \frac{1}{2}$

Step 5: Solve for $\sin^2 20$

Multiply through by $\cos 20$: $\sin^2 20 = \frac{1}{2} \cdot \cos 20$

Thus, the equation simplifies to: $\sin^2 20 = \frac{\cos 20}{2}$

If you have further questions or need more clarifications, feel free to ask!

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Related Questions:

1. What is the value of $\sin^2 20 + \cos^2 20$?
2. How does the double-angle identity relate to $\sin 20$ and $\cos 20$?
3. Can $\tan 20$ be simplified into a known trigonometric expression?
4. How would this equation change if $20^\circ$ is replaced by $x$?
5. What are the approximate values of $\sin 20$ and $\cos 20$?

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Tip:

Always check for trigonometric identities like Pythagoras ($\sin^2 x + \cos^2 x = 1$) to simplify equations effectively.