The simplification of 1+tan^{2} x is equal to sec^{2} x. In this post, we will find the formula of 1+tan^{2}x. This will be proved using trigonometric identities.

## 1+tan^{2}x Formula

To simplify the expression 1+tan^{2} x, we will follow the below steps:

**Step 1:** At first, we put $\tan x=\dfrac{\sin x}{\cos x}$. By doing so, we obtain that

$1+\tan^2 x$ $=1+(\dfrac{\sin x}{\cos x})^2$

$=1+\dfrac{\sin^2 x}{\cos^2 x}$

**Step 2:** Simplifying the above, we get that

$1+\tan^2 x=1+\dfrac{\sin^2 x}{\cos^2 x}$

$=\dfrac{\sin^2 x+\cos^2 x}{\cos^2 x}$

$=\dfrac{1}{\cos^2 x}$ as we know that $\sin^2 x+\cos^2 x=1$

$=(\dfrac{1}{\cos x})^2$ $=(\sec x)^2$ as the reciprocal of cosx is secx.

$=\sec^2 x$

**Conclusion:** Thus, the simplification of $1+\tan^2 x$ is equal to $\sec^2 x$.

**Note** that we also have that $1+\tan^2 \theta=\sec^2 \theta$ obtained in a similar way as above.

**Also Read:**

**sinx=0, cosx=0, tanx=0 General Solution**

**Values of sin 15, cos 15, tan 15**

**Values of sin 75, cos 75, tan 75**

## Question-Answer on 1+tan^{2}x Formula

**Question 1:** Find the value of 1+tan^{2} $45^\circ$

*Answer: *

From the above, we get that the value of 1+tan^{2} x is equal to sec^{2} x. In this formula, we put $x=45^\circ$. Thus, we have

$1+\tan^2 45^\circ$ $=\sec^2 45^\circ$

$=\dfrac{1}{\cos^2 45^\circ}$

$=\dfrac{1}{(\dfrac{1}{\sqrt{2}})^2}$ as we know that $\cos 45^\circ =\dfrac{1}{\sqrt{2}}$.

$=\dfrac{1}{1/2}=2$

Thus, the value of $1+\tan^2 45^\circ$ is equal to $2$.

## FAQs

**Q1: What is the formula of 1+tan ^{2}x?**

**Answer:** The formula of 1+tan^{2}x is given as follows: 1+tan^{2}x = sec^{2}x.