1-sec^2x Formula, Identity | 1-sec^2 theta is equal to

The simplification of the expression 1-sec2x is equal to -tan2x. In this post, we will establish 1-sec^2x formula.

We have:

  • 1- sec2x = – tan2x
  • 1- sec2θ = – tan2θ.
Simplify 1-sec^2x

Let us now prove the above formulas.

1-sec2x Formula

The formula of 1-sec2x is given below:

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

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-sec2x:

  1. sin2x +cos2x = 1
  2. secx = 1/cosx 
  3. tanx = sinx/cosx

Step 2: Substituting the value of secx from the above in the expression 1-sec2x, we will get that

1-sec2x = $1-\left(\dfrac{1}{\cos x} \right)^2$

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

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

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

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

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

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

= – tan2x

Thus, the simplification or the formula of 1-sec2x is equal to – tan2x.

Remark: Putting x=θ in the above formula, we get the following simplification of 1-sec2θ:

1-sec2θ = – tan2θ.

Also Read:

Formula of 1+tan^2 x

Simplify cos(x-pi)

Simplify 1-sin2x

Simplify 1-cos2x

Question-Answer on 1-sec2x Formula

Question 1: Find the value of 1-sec245°


From the above, we know that

1-sec2θ = – tan2θ

Put θ = 45°.

So we get that

1-sec245° = – tan245°

= -12 as we know that tan45°=1.

= -1

So the value of 1-sec245° is equal to -1.


Q1: What is the formula of 1-sec2x?

Answer: The formula of 1-sec2x is given by 1-sec2x= -tan2x.

Q2: What is the formula of 1-sec2θ?

Answer: The formula of 1-sec2θ is given as follows: 1-sec2θ= -tan2θ.

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