The simplification of the expression 1-sec^{2}x is equal to -tan^{2}x. In this post, we will establish 1-sec^2x formula.

We have:

- 1- sec
^{2}x = – tan^{2}x - 1- sec
^{2}θ = – tan^{2}θ.

Let us now prove the above formulas.

## 1-sec^{2}x Formula

The formula of 1-sec^{2}x is given below:

$\boxed{1-\sec^2 x=-\tan^2 x}$

**Proof:**

To simplify the expression $1-\sec^2 x$, we will follow the below steps:

**Step 1:** We will use the following trigonometric identities to find the value of 1-sec^{2}x:

- sin
^{2}x +cos^{2}x = 1 - secx = 1/cosx
- tanx = sinx/cosx

**Step 2:** Substituting the value of secx from the above in the expression 1-sec^{2}x, we will get that

1-sec^{2}x = $1-\left(\dfrac{1}{\cos x} \right)^2$

= $1-\dfrac{1}{\cos^2 x}$

= $\dfrac{\cos^2 x -1}{\cos^2 x}$

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

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

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

= $- \left(\dfrac{\sin x}{\cos x} \right)^2$

= – tan^{2}x

Thus, the simplification or the formula of 1-sec^{2}x is equal to – tan^{2}x.

**Remark:** Putting x=θ in the above formula, we get the following simplification of 1-sec^{2}θ:

1-sec^{2}θ = – tan^{2}θ.

**Also Read:**

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

**Question 1:** Find the value of 1-sec^{2}45°

*Answer:*

From the above, we know that

1-sec^{2}θ = – tan^{2}θ

Put θ = 45°.

So we get that

1-sec^{2}45° = – tan^{2}45°

= -1^{2 }as we know that tan45°=1.

= -1

So the value of 1-sec^{2}45° is equal to -1.

## FAQs

**Q1: What is the formula of 1-sec ^{2}x?**

**Answer:** The formula of 1-sec^{2}x is given by 1-sec^{2}x= -tan^{2}x**.**

**Q2: What is the formula of 1-sec ^{2}θ?**

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