Question
\( \begin{array}{lll}16.4 & \tan \alpha+\frac{1}{\tan \alpha}=\frac{2}{\sin 2 \alpha} & \begin{array}{l}\text { (5) } \\ \text { L3 } \\ 16.5 \\ \cos 4 x=1-8 \sin ^{2} x+8 \sin ^{4} x\end{array} \\ 16.6 & \cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta & \text { (4) } \\ \text { L4 } \\ \text { (4) } & \text { L3 }\end{array} \)
Ask by Page Todd.
Feb 23,2025 11:29
UpStudy AI Solution
Tutor-Verified Answer
Answer
Here are the simplified proofs for the given trigonometric identities:
1. **Identity:** tan α + (1/tan α) = 2/sin 2α
- tan α + cot α = 1/(sin α cos α)
- Using sin 2α = 2 sin α cos α, we get 2/sin 2α.
2. **Identity:** cos 4x = 1 – 8 sin²x + 8 sin⁴x
- cos 4x = 1 – 8 sin²x cos²x
- Substitute cos²x = 1 – sin²x to get 1 – 8 sin²x + 8 sin⁴x.
3. **Identity:** cos 3θ = 4 cos³θ – 3 cos θ
- Use the triple-angle formula: cos 3θ = 4 cos³θ – 3 cos θ.
These identities are useful for simplifying trigonometric expressions and solving equations.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
Bonus Knowledge
In the exploration of trigonometric identities, it’s fascinating to note how these relationships have been used throughout history, especially in astronomy and navigation. Ancient Greeks employed trigonometric functions to calculate distances and angles in their vast explorations, paving the way for modern mathematics and physics!
When tackling trigonometric problems, a common mistake is neglecting to convert between degrees and radians, which can lead to incorrect answers. Always double-check which unit you're working with! Another helpful tip is to sketch the unit circle; it can help visualize angles and their sine and cosine values, making complex transformations a breeze.
Related Questions
Please type the verification code shown:
