Question

The value of $\tan 48^\circ \tan 16^\circ \tan 42^\circ \tan 74^\circ$ is _____.

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $2$

Hint:

Whenever trigonometric angles add up to \(90^\circ\), immediately think of complementary identities: \[ \tan(90^\circ-\theta)=\cot\theta \] \[ \sin(90^\circ-\theta)=\cos\theta \] These identities help simplify lengthy expressions very quickly.

Answer 1

The Correct Option is B

Concept: This problem is based on complementary angles and the trigonometric identity: \[ \tan(90^\circ-\theta)=\cot\theta \] We also use another important identity: \[ \tan\theta \cdot \cot\theta = 1 \] Whenever angles add up to \(90^\circ\), their tangent values become reciprocals of each other.

Step 1: Observe the given angles carefully.
The expression is: \[ \tan 48^\circ \tan 16^\circ \tan 42^\circ \tan 74^\circ \] Now check which angles are complementary. We notice: \[ 48^\circ + 42^\circ = 90^\circ \] and \[ 16^\circ + 74^\circ = 90^\circ \] Thus:
• \(42^\circ\) is complementary to \(48^\circ\)
• \(74^\circ\) is complementary to \(16^\circ\)

Step 2: Convert complementary tangent functions into cotangent functions.
Using: \[ \tan(90^\circ-\theta)=\cot\theta \] we get: \[ \tan 42^\circ = \tan(90^\circ-48^\circ)=\cot48^\circ \] Similarly, \[ \tan74^\circ=\tan(90^\circ-16^\circ)=\cot16^\circ \] Now substitute these into the expression. \[ \tan48^\circ \tan16^\circ \cot48^\circ \cot16^\circ \]

Step 3: Rearrange the terms for simplification.
Group corresponding tangent and cotangent pairs: \[ (\tan48^\circ \cdot \cot48^\circ) (\tan16^\circ \cdot \cot16^\circ) \]

Step 4: Apply the identity.
We know: \[ \tan\theta \cdot \cot\theta = 1 \] Therefore, \[ \tan48^\circ \cdot \cot48^\circ = 1 \] and \[ \tan16^\circ \cdot \cot16^\circ = 1 \] Hence, \[ 1 \times 1 = 1 \] Therefore, \[ \boxed{1} \] is the required value.