Sure! Let's work through this problem step-by-step to verify the given trigonometric identity:
### Step 1: Express [tex]\(\sec A\)[/tex], [tex]\(\cos A\)[/tex], [tex]\(\tan A\)[/tex], and [tex]\(\sin A\)[/tex] in terms of [tex]\(A\)[/tex]
From trigonometry, we know the following identities:
- [tex]\(\sec A = \frac{1}{\cos A}\)[/tex]
- [tex]\(\tan A = \frac{\sin A}{\cos A}\)[/tex]
### Step 2: Calculate [tex]\((\sec A + \cos A)(\sec A - \cos A)\)[/tex]
This expression can be treated as a difference of squares. We can apply the formula [tex]\((x + y)(x - y) = x^2 - y^2\)[/tex]:
[tex]\[
(\sec A + \cos A)(\sec A - \cos A) = \sec^2 A - \cos^2 A
\][/tex]
### Step 3: Substitute [tex]\(\sec^2 A\)[/tex] with the identity involving [tex]\(\tan A\)[/tex]
Recall the Pythagorean identity for secant and tangent:
[tex]\[
\sec^2 A = 1 + \tan^2 A
\][/tex]
Using this identity, we can rewrite the expression:
[tex]\[
\sec^2 A - \cos^2 A = (1 + \tan^2 A) - \cos^2 A
\][/tex]
### Step 4: Rewrite [tex]\(\cos^2 A\)[/tex] as [tex]\(1 - \sin^2 A\)[/tex]
We also know from the Pythagorean identity for sine and cosine:
[tex]\[
\cos^2 A = 1 - \sin^2 A
\][/tex]
Thus, we replace [tex]\(\cos^2 A\)[/tex] in the equation:
[tex]\[
(1 + \tan^2 A) - (1 - \sin^2 A)
\][/tex]
### Step 5: Simplify the equation
Now, simplify the equation step by step:
[tex]\[
(1 + \tan^2 A) - 1 + \sin^2 A = \tan^2 A + \sin^2 A
\][/tex]
### Conclusion
We find that the left-hand side of the equation [tex]\((\sec A + \cos A)(\sec A - \cos A)\)[/tex] simplifies to [tex]\(\tan^2 A + \sin^2 A\)[/tex], which equals the right-hand side of the given identity. The equality holds true.
### Verification with specific values
With the given angle [tex]\(A = 45^\circ\)[/tex] (or [tex]\(\frac{\pi}{4}\)[/tex] radians):
- [tex]\(\sec A = 1.414213562373095\)[/tex]
- [tex]\(\cos A = 0.7071067811865476\)[/tex]
- [tex]\(\tan A = 0.9999999999999999\)[/tex]
- [tex]\(\sin A = 0.7071067811865475\)[/tex]
Calculating both sides:
- Left-hand side: [tex]\((\sec A + \cos A)(\sec A - \cos A) = 1.4999999999999996\)[/tex]
- Right-hand side: [tex]\(\tan^2 A + \sin^2 A = 1.4999999999999996\)[/tex]
Since both sides are equal, the identity is verified true for [tex]\(A = 45^\circ\)[/tex].
So, the trigonometric identity holds true:
[tex]\[
(\sec A + \cos A)(\sec A - \cos A) = \tan^2 A + \sin^2 A
\][/tex]
