Proving Trigonometric Identities | Fundamental Trigonometric Identities | Formulas | Sample Problems

Summary

This educational video offers a comprehensive guide to proving trigonometric identities. It begins with reciprocal identities, such as sine and secant, and progresses to quotient identities, like tangent and cotangent. The video then delves into Pythagorean identities, illustrating how to derive them from basic trigonometric functions. Three examples are provided, demonstrating the process of proving identities step-by-step, including simplifying expressions and applying fundamental trigonometric relationships. The host, Prof D, encourages viewers to engage with the content by asking questions in the comments section.

Takeaways

• 📚 The video is a tutorial on proving trigonometric identities, focusing on fundamental identities.
• 🔄 The presenter introduces reciprocal identities, such as \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• 🔢 The video covers quotient identities, including \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
• 📐 Pythagorean identities are discussed, like \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and \( \tan^2(\theta) + 1 = \sec^2(\theta) \).
• 📝 The presenter demonstrates how to simplify left and right sides of equations using trigonometric identities.
• 👨‍🏫 Example proofs are provided, such as proving \( \sin(\theta) \times \sec(\theta) = \tan(\theta) \).
• 🧩 The video emphasizes the importance of converting complex expressions into simpler forms using identities.
• 🔎 The presenter shows how to apply identities to prove more complex relationships, like \( \sec^2(\theta) - \cos^2(\theta) = \tan^2(\theta) - \cot^2(\theta) \).
• 🤓 The video is educational, aiming to help viewers understand and prove trigonometric identities.
• 👋 The presenter invites viewers to ask questions or seek clarifications in the comments section.

Q & A

What are the reciprocal identities mentioned in the script?

The reciprocal identities mentioned are: secant theta (sec(θ)) is equal to 1/cosine theta (1/cos(θ)), and cotangent theta (cot(θ)) is equivalent to 1/tangent theta (1/tan(θ)).

What is the relationship between sine and secant according to the script?

According to the script, secant theta (sec(θ)) is the reciprocal of cosine theta (cos(θ)), and it can be expressed as 1/cos(θ), which is also equal to 1/(1/sin(θ)), making sec(θ) equal to 1/sin(θ).

How is tangent theta related to secant and cosine theta?

The script explains that tangent theta (tan(θ)) is equal to sine theta (sin(θ)) divided by cosine theta (cos(θ)), which is also the reciprocal of secant theta (sec(θ)).

What is the quotient identity for tangent theta?

The quotient identity for tangent theta (tan(θ)) is sin(θ)/cos(θ), which is also equal to the reciprocal of cotangent theta (cot(θ)).

What is the Pythagorean identity for sine and cosine?

The Pythagorean identity for sine and cosine is sin²(θ) + cos²(θ) = 1, which is derived from the Pythagorean theorem applied to a right triangle.

How is the identity for tangent squared theta plus 1 related to secant squared theta?

The identity states that tan²(θ) + 1 = sec²(θ). This means that the square of the tangent of an angle plus one is equal to the square of the secant of that angle.

What is the relationship between 1 plus cotangent squared theta and cosecant squared theta?

The script mentions that 1 + cot²(θ) = csc²(θ). This identity shows that one plus the square of the cotangent of an angle is equal to the square of the cosecant of that angle.

How does the script prove that sine theta times secant theta equals tangent theta?

The script proves this by converting secant theta to its reciprocal form in terms of sine and cosine, resulting in sin(θ)/cos(θ), which is the definition of tangent theta (tan(θ)).

What is the proof that one over sine theta times cotangent theta equals one over cosine theta?

The proof involves expressing cotangent theta in terms of sine and cosine, resulting in cos(θ)/sin(θ), which when multiplied by 1/sin(θ) gives 1/cos(θ), matching the right side of the equation.

How is the identity secant squared theta minus cosine squared theta equal to tangent squared theta minus cotangent squared theta proven?

The script uses the identity sec²(θ) = 1 + tan²(θ) and rearranges it to sec²(θ) - 1 = tan²(θ), then substitutes cotangent squared theta as the negative of cosecant squared theta minus one, resulting in the left side equaling the right side of the identity.