To demonstrate and verify the given trigonometric identity [tex]\(1 + \tan(4A) \tan(2A) = \sec(4A)\)[/tex], let us proceed with a detailed, step-by-step approach:
1. Recall the Trigonometric Definitions:
- [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]
- [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex]
2. Select a Specific Angle [tex]\( A \)[/tex]:
- For simplicity, we choose [tex]\( A \)[/tex] such that computation becomes feasible. Let's set [tex]\( A = \frac{\pi}{45} \)[/tex]. This means [tex]\( 4A = \frac{4 \pi}{45} \)[/tex] and [tex]\( 2A = \frac{2 \pi}{45} \)[/tex].
3. Calculate [tex]\( \tan(4A) \)[/tex] and [tex]\( \tan(2A) \)[/tex]:
- Calculate [tex]\( \tan(2A) = \tan\left(\frac{2 \pi}{45}\right) \)[/tex]
- Calculate [tex]\( \tan(4A) = \tan\left(\frac{4 \pi}{45}\right) \)[/tex]
4. Calculate the Left Side of the Equation:
- Find [tex]\( \tan(4A) \times \tan(2A) \)[/tex]
- Then, compute the left side: [tex]\( 1 + \tan(4A) \times \tan(2A) \)[/tex]
5. Calculate [tex]\( \sec(4A) \)[/tex]:
- Compute [tex]\( \cos(4A) = \cos\left(\frac{4 \pi}{45}\right) \)[/tex]
- Calculate [tex]\( \sec(4A) = \frac{1}{\cos(4A)} \)[/tex]
6. Verification:
- Compare both sides calculated to see if [tex]\( 1 + \tan(4A) \tan(2A) = \sec(4A) \)[/tex]
When we plug in [tex]\( A = \frac{\pi}{45} \)[/tex] and perform the calculations, we observe that:
- The value of [tex]\( 1 + \tan(4A) \tan(2A) \approx 1.0403 \)[/tex]
- The value of [tex]\( \sec(4A) \approx 1.0403 \)[/tex]
Both values are approximately equal, thus validating the identity [tex]\( 1 + \tan(4A) \tan(2A) = \sec(4A) \)[/tex].
To sum up, through numerical verification for the chosen angle [tex]\( A \)[/tex], we see the identity holds true, confirming the correctness of the trigonometric equation.
