Solution:

Given,

tan θ = \[12/13\] …….. \[\left( 1 \right)\]

We know that by definition,

tan θ = Perpendicular side opposite to ∠θ / Base side adjacent to ∠θ …… \[\left( 2 \right)\]

On comparing equation \[\left( 1 \right)\] and \[\left( 2 \right),\] we have

Perpendicular side opposite to ∠θ = \[12\]

Base side adjacent to ∠θ = \[13\]

Thus, in the triangle representing ∠ θ we have,

Hypotenuse AC is the unknown and it can be found by using Pythagoras theorem

So by applying Pythagoras theorem, we have

AC² = \[{{12}^{2}}~+\text{ }{{13}^{2}}\]

AC ² = \[144\text{ }+\text{ }169\]

AC² = \[313\pi \]

⇒ AC = \[\surd 313\]

By definition,

sin θ = Perpendicular side opposite to ∠θ / Hypotenuse = AB / AC

⇒ sin θ = \[12/\text{ }\surd 313\ldots ..\left( 3 \right)\]

And, cos θ = Base side adjacent to ∠θ / Hypotenuse = BC / AC

⇒ cos θ = \[13/\text{ }\surd 313\text{ }\ldots ..\left( 4 \right)\]

Now, substituting the value of sin θ and cos θ from equation \[\left( 3 \right)\] and \[\left( 4 \right)\] respectively in the equation below

Therefore,