Question

The value of is \(\frac{tan 65°}{ cot 25°}\) is

• A. 0
• B. 1
• C. -1
• D. \(\sqrt3\)

Answer 1

The Correct Option is B

Step 1: Use the identity for cotangent.

The cotangent of an angle is the reciprocal of the tangent:

\[ \cot 25^\circ = \frac{1}{\tan 25^\circ} \]

Substitute this into the given expression:

\[ \frac{\tan 65^\circ}{\cot 25^\circ} = \frac{\tan 65^\circ}{\frac{1}{\tan 25^\circ}} \]

Simplify the division:

\[ \frac{\tan 65^\circ}{\cot 25^\circ} = \tan 65^\circ \cdot \tan 25^\circ \]

Step 2: Use the complementary angle identity.

Since \( 65^\circ + 25^\circ = 90^\circ \), we know that:

\[ \tan 65^\circ = \cot 25^\circ \]

Thus, the expression becomes:

\[ \tan 65^\circ \cdot \tan 25^\circ = \cot 25^\circ \cdot \tan 25^\circ \]

Using the identity \( \cot x \cdot \tan x = 1 \):

\[ \cot 25^\circ \cdot \tan 25^\circ = 1 \]

Final Answer: The value of \( \frac{\tan 65^\circ}{\cot 25^\circ} \) is \( \mathbf{1} \).