Question

If \(\tan 25^\circ \times \tan 65^\circ = \sin A\) then the value of A is

• A. 25°
• B. 65°
• C. 90°
• D. 45°

Hint:

A useful property to remember is: if A + B = 90°, then \(\tan A \times \tan B = 1\). This allows you to simplify the left side of the equation in a single step.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires simplifying the product of two tangent functions using complementary angle identities, and then solving the resulting trigonometric equation.

Step 2: Key Formula or Approach:
The key identities are:
\[ \tan(90^\circ - \theta) = \cot \theta \] \[ \tan \theta \times \cot \theta = 1 \]

Step 3: Detailed Explanation:
First, let's simplify the left side of the equation: \(\tan 25^\circ \times \tan 65^\circ\).
Notice that \(25^\circ + 65^\circ = 90^\circ\), so the angles are complementary.
We can rewrite \(\tan 65^\circ\) using the identity:
\[ \tan 65^\circ = \tan(90^\circ - 25^\circ) = \cot 25^\circ \] Now substitute this back into the expression:
\[ \tan 25^\circ \times \cot 25^\circ \] Since \(\cot \theta = \frac{1}{\tan \theta}\), the product is:
\[ \tan 25^\circ \times \frac{1}{\tan 25^\circ} = 1 \] So, the original equation becomes:
\[ 1 = \sin A \] We need to find the angle A for which \(\sin A = 1\).
From standard trigonometric values, we know that \(\sin 90^\circ = 1\).
Therefore, \(A = 90^\circ\).

Step 4: Final Answer:
The value of A is 90°.