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Q.

Is it true that tan 48° tan 23° tan 42° tan 67° = 1?


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a

True

b

False 

answer is A.

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Detailed Solution

Given the equation, tan 48° tan 23° tan 42° tan 67° = 1.
Let us consider the LHS = tan 48° tan 23° tan 42° tan 67°.
We can write tan in terms of cot by using the formula cot θ = tan(90°- θ).
Thus, tan 48° = tan (90° – 42°)
∴ tan 48°= cot 42°
tan 23° = tan (90° – 67°)
∴ tan23° = cot 67°
Substituting the values of tan 48° and tan23° in the above expression, we get,
⇒LHS = cot 42° cot 67° tan 42° tan 67°
⇒LHS = (cot 42° tan 42°) (cot 67° tan 67°)
Since cot θ.tan θ = 1
⇒LHS = 1×1
⇒LHS = 1
∴ LHS = RHS
Thus the given statement tan 48° tan 23° tan 42° tan 67° = 1 is true.
Therefore, the correct answer is option (1).
 
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