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.

Let us now prove the above formulas.

1-sec²x Formula

The formula of 1-sec²x is given below:

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

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-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°

Answer:

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.

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