Does the following trigonometric equation have any solutions? If Yes, obtain the solution(s):

OR

Determine whether the operation * define below on Q is binary operation or not.

a * b = ab+1

If yes, check the commutative and the associative properties. Also check the existence of identity element and the inverse of all elements in.

We know,

Now,

[∵ tan^-1(-x) = -tan^-1x]

⇒ -7 + 7x = 2x² – 2x + 2

⇒ 2x² – 9x + 9 = 0

⇒ 2x² – 6x – 3x + 9 = 0

⇒ 2x(x – 3) – 3(x – 3) = 0

⇒ (2x – 3)(x – 3) = 0

OR

a*b = ab + 1,

Let p, q ∈ Q,

p*q = pq + 1 ∈ Q

As, P*Q also belong to Q, * defined on Q is binary operation.

Commutative:

Let p, q ∈ Q,

p*q = pq + 1

q*p = qp + 1

As,

pq = qp

⇒ pq + 1 = qp + 1

⇒ p*q = q*p

∴ * satisfies commutative property!

Associativity:

Let p, q, r ∈ Q,

Here,

(p*q)*r = (pq + 1)*r

= (pq + 1)r + 1

= pqr + r + 1

and

p*(q*r) = p*(qr + 1)

= p(qr + 1) + 1

= pqr + p + 1

⇒ (p*q)*r = p*(q*r)

∴ (*) doesn’t satisfy associative property!

Identity:

Let identity be e, then p*e = p

⇒ pe + 1 = p

⇒ pe = p – 1

As, identity is not unique, the binary operation don’t have a identity and as identity is not there, inverse is absurd!