Question

Value of:

$Tan 20 \cdot Cot 50 \cdot Tan 80 = ?$

Answer 1

$\sqrt{3}$

Answer 2

To find the value of the expression $Tan 20^\circ \cdot Cot 50^\circ \cdot Tan 80^\circ$, we will use trigonometric identities.

Step 1: Convert $Cot 50^\circ$ to an equivalent $Tan$ form. We know that $Cot \theta = Tan (90^\circ - \theta)$. So, $Cot 50^\circ = Tan (90^\circ - 50^\circ) = Tan 40^\circ$.

Step 2: Substitute the converted term back into the expression. The expression becomes: $Tan 20^\circ \cdot Tan 40^\circ \cdot Tan 80^\circ$

Step 3: Recognize the pattern. This expression fits the general trigonometric identity: $Tan \theta \cdot Tan (60^\circ - \theta) \cdot Tan (60^\circ + \theta) = Tan (3\theta)$

In our expression, if we let $\theta = 20^\circ$:

• The first term is $Tan 20^\circ$.
• The second term is $Tan 40^\circ = Tan (60^\circ - 20^\circ)$.
• The third term is $Tan 80^\circ = Tan (60^\circ + 20^\circ)$.

Thus, the expression exactly matches the pattern with $\theta = 20^\circ$.

Step 4: Apply the identity. Using the identity, the value of the expression is $Tan (3 \cdot \theta)$. Substitute $\theta = 20^\circ$: $Tan (3 \cdot 20^\circ) = Tan 60^\circ$

Step 5: Recall the standard value of $Tan 60^\circ$. We know that $Tan 60^\circ = \sqrt{3}$.

Therefore, the value of the given expression is $\sqrt{3}$.