Find the value of

sin A. cos B + cos A. sin B. if and tan B = √3.

Given:

Let,

Side opposite to angle A =BC = 1k

Side adjacent to angle A =AB = k√3

where, k is any positive integer

Firstly we have to find the value of BC.

So, we can find the value of AC with the help of Pythagoras theorem

⇒ (AB)² + (BC)² = (AC)²

⇒ (√3k)² + (1k)² = (AC)²

⇒ (AC)² = 1 k² +3 k²

⇒ (AC)² = 4 k²

⇒ AC =√2 k²

⇒ AC =±2k

But side AC can’t be negative. So, AC = 2k

Now, we will find the sin A and cos A

Side opposite to angle A = BC = k

and Hypotenuse = AC = 2k

So,

Now, we know that,

Side adjacent to angle A = AB =k√3

Hypotenuse = AC =2k

So,

Now,

Given: tan B = √3

Let,

Side opposite to angle B =AC = √3k

Side adjacent to angle B =AB = 1k

where, k is any positive integer

Firstly we have to find the value of BC.

So, we can find the value of AC with the help of Pythagoras theorem

⇒ (AB)² + (AC)² = (BC)²

⇒ (1k)² + (√3k)² = (BC)²

⇒ (BC)² = 1 k² +3 k²

⇒ (BC)² = 4 k²

⇒ BC =√2 k²

⇒ BC =±2k

But side BC can’t be negative. So, BC = 2k

Now, we will find the sin B and cos B

Side opposite to angle B = AC = k√3

and Hypotenuse = BC = 2k

So,

Now, we know that,

Side adjacent to angle B = AB =1k

Hypotenuse = BC =2k

So,

Now, sin A. cos B + cos A. sin B

Putting the values of sin A, sin B cos A and Cos B, we get

=1