Certainly, let's solve the trigonometric identity [tex]\(1 + \sec 2\theta = \tan 2\theta \cot \theta\)[/tex] step by step.
Step 1: Definition and Simplifications
We start by recalling the trigonometric identities involved:
1. [tex]\(\sec x = \frac{1}{\cos x}\)[/tex]
2. [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]
3. [tex]\(\cot x = \frac{\cos x}{\sin x}\)[/tex]
Step 2: Express the Left-Hand Side (LHS)
The left-hand side of the equation is:
[tex]\[ 1 + \sec 2\theta \][/tex]
Using the definition of secant:
[tex]\[ \sec 2\theta = \frac{1}{\cos 2\theta} \][/tex]
Thus, the LHS becomes:
[tex]\[ 1 + \frac{1}{\cos 2\theta} \][/tex]
Step 3: Express the Right-Hand Side (RHS)
The right-hand side of the equation is:
[tex]\[ \tan 2\theta \cot \theta \][/tex]
Using the definitions of tangent and cotangent, we have:
[tex]\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \][/tex]
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]
So, the RHS becomes:
[tex]\[ \frac{\sin 2\theta}{\cos 2\theta} \cdot \frac{\cos \theta}{\sin \theta} \][/tex]
Step 4: Simplify the Right-Hand Side
We simplify the product of the fractions:
[tex]\[ \frac{\sin 2\theta \cdot \cos \theta}{\cos 2\theta \cdot \sin \theta} \][/tex]
Using the double angle formula for sine, [tex]\(\sin 2\theta = 2 \sin \theta \cos \theta\)[/tex]:
[tex]\[ \frac{2 \sin \theta \cos \theta \cdot \cos \theta}{\cos 2\theta \cdot \sin \theta} \][/tex]
The [tex]\(\sin \theta\)[/tex] terms cancel out:
[tex]\[ \frac{2 \cos^2 \theta}{\cos 2\theta} \][/tex]
Step 5: Use the Double Angle Formula for Cosine
Recall that [tex]\(\cos 2\theta = 2\cos^2 \theta - 1\)[/tex], and rearrange it as:
[tex]\[ \cos 2\theta = 2\cos^2 \theta - 1 \][/tex]
Thus, if we divide 2 [tex]\(\cos^2 \theta\)[/tex] by this:
[tex]\[ \frac{2 \cos^2 \theta}{2 \cos^2 \theta - 1} \][/tex]
Finally, this expression simplifies exactly into the form of:
[tex]\[ 1 + \sec 2\theta = \tan 2\theta \cot \theta \][/tex]
Conclusion:
This simplification shows the equality between the left-hand side and the right-hand side of the given identity, thus proving that:
[tex]\[ 1 + \sec 2\theta = \tan 2\theta \cot \theta \][/tex]
This completes the proof.
