To prove the identity
[tex]\[
\left(\sec^2 x - 1\right) \cos x = \tan x \sin x
\][/tex]
we will use trigonometric identities and algebraic manipulations step-by-step.
### Step 1: Start with the Left-Hand Side (LHS)
The LHS of the given identity is:
[tex]\[
\left(\sec^2 x - 1\right) \cos x
\][/tex]
### Step 2: Use the trigonometric identity
We know from trigonometric identities that:
[tex]\[
\sec^2 x - 1 = \tan^2 x
\][/tex]
Substituting this identity into the LHS, we get:
[tex]\[
\left(\tan^2 x\right) \cos x
\][/tex]
### Step 3: Simplify the expression
Recall the definition of [tex]\(\tan x\)[/tex] as:
[tex]\[
\tan x = \frac{\sin x}{\cos x}
\][/tex]
So, [tex]\(\tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2 = \frac{\sin^2 x}{\cos^2 x}\)[/tex].
Substituting [tex]\(\tan^2 x\)[/tex] back, we obtain:
[tex]\[
\frac{\sin^2 x}{\cos^2 x} \cos x = \frac{\sin^2 x}{\cos x}
\][/tex]
### Step 4: Simplify further
Now, simplify the expression:
[tex]\[
\frac{\sin^2 x}{\cos x} = \sin x \cdot \frac{\sin x}{\cos x} = \sin x \tan x
\][/tex]
Thus, we have:
[tex]\[
\left(\sec^2 x - 1\right) \cos x = \sin x \tan x
\][/tex]
### Step 5: Compare with the Right-Hand Side (RHS)
The RHS of the given identity is:
[tex]\[
\tan x \sin x
\][/tex]
Clearly,
[tex]\[
\sin x \tan x = \tan x \sin x
\][/tex]
### Conclusion
Both sides are equal:
[tex]\[
\left(\sec^2 x - 1\right) \cos x = \tan x \sin x
\][/tex]
Hence, the identity is proven to be true.
