Derive the expression 1 + tan²x.

In a right-angled triangle, the tangent of an angle is defined as 'The ratio of the length of the side opposite the angle to the length of the adjacent side'.

Answer: The expression 1 + tan²x can be expressed as 1 + tan²x = sec²x

Let us proceed step by step to get the required expression.

Explanation:

We can proceed by using sine and cosine identity as 

 sin²x + cos²x = 1 ——- (i)

Dividing both sides of the equation by cos²x

We get,

(sin²x / cos²x) + (cos²x / cos²x) = 1/cos²x

From trigonometric identity, we already know that

sin²x / cos² x = tan²x , and cos²x / cos²x = 1

On substituting both the values in eqn (i), we get

tan²x + 1= 1/ cos²x ——–(ii)

Now we know that, 1/ cos² x = sec²x

On replacing 1/ cos²x with sec²x, equation (ii) becomes tan²x + 1 = sec²x

The expression 1 + tan²x can be expressed as tan²x + 1 = sec²x