# Derive the expression 1 + tan^{2}x.

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 + tan^{2}x can be expressed as 1 + tan^{2}x = sec^{2}x

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

**Explanation:**

We can proceed by using sine and cosine identity as

sin^{2}x + cos^{2}x = 1 ——- (i)

Dividing both sides of the equation by cos^{2}x

We get,

(sin^{2}x / cos^{2}x) + (cos^{2}x / cos^{2}x) = 1/cos^{2}x

From trigonometric identity, we already know that

sin^{2}x / cos^{2 }x = tan^{2}x , and cos^{2}x / cos^{2}x = 1

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

tan^{2}x + 1= 1/ cos^{2}x ——–(ii)

Now we know that, 1/ cos^{2 }x = sec^{2}x

On replacing 1/ cos^{2}x with sec^{2}x, equation (ii) becomes tan^{2}x + 1 = sec^{2}x

### The expression 1 + tan^{2}x can be expressed as tan^{2}x + 1 = sec^{2}x

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