Let P be the relation defined on the set of all real numbers such that P = a , b : sec 2 a - tan 2 b = 1 .…

Let P be the relation defined on the set of all real numbers such that P=a,b:sec2a-tan2b=1. Then, P is
  1. reflexive and symmetric but not transitive
  2. symmetric and transitive but not reflexive
  3. reflexive and transitive but not symmetric
  4. an equivalence relation

Solution

P=a,b:sec2a-tan2b=1

P=a,a:sec2a-tan2a=1

its reflexive relation.

Now P=a,b:sec2a-tan2b=1

sec2a-tan2b=1

⇒sec2a=1+tan2b

⇒sec2a=sec2b

⇒sec a=sec b

sec2a-tan2b=1

1+tan2a-sec2b+1=1

sec2b-tan2a=1

Hence, it is symmetric.

If sec a=sec b and sec b=sec c then sec a=sec c⇒ transitive 

So, it is an equivalence relation.

Asked in: JEE Main 2014 (09 Apr Online)

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