Trigonometric Identities and Proofs

Anil Kumar presents a series on trigonometric identities, focusing on proving three key identities. The video begins with an introduction to basic trigonometric Pythagorean identities, followed by detailed step-by-step proofs of each identity. The first identity involves simplifying expressions to show equality, the second uses difference of squares, and the third explores multiple methods to reach the solution. The video emphasizes understanding and applying standard trigonometric formulas to prove identities effectively.

What is the basic Pythagorean identity involving sine and cosine?

• sin(x) - cos(x) = 1

• sin^2(x) + cos^2(x) = 1

• sin^2(x) - cos^2(x) = 1

• sin(x) + cos(x) = 1

Which identity is equivalent to 1 + tan^2(x)?

• sin^2(x)

• sec^2(x)

• csc^2(x)

• cot^2(x)

How is tan(x) expressed in terms of sine and cosine?

• 1 / cos(x)

• 1 / sin(x)

• cos(x) / sin(x)

• sin(x) / cos(x)

In proving the first identity, what is the common denominator used?

• 1 - sin(x) times cos(x)

• 1 + sin(x) times cos(x)

• 1 - cos(x) times sin(x)

• sin(x) times cos(x)

What is the result of simplifying cos^2(x) - 1 - sin(x)?

• 1 - sin^2(x)

• 1 + sin(x)

• sin(x) - sin^2(x)

• cos(x) - sin(x)

What is the final expression obtained in the proof of the first identity?

• csc(x)

• cot(x)

• sec(x)

• tan(x)

In the second identity, how is sec(x) expressed?

• sin(x) / cos(x)

• 1 / cos(x)

• 1 / sin(x)

• cos(x) / sin(x)

What is the factorization of 1 - cos^2(x)?

• (1 + cos(x))(1 - cos(x))

• (1 - cos(x))(1 + sin(x))

• (1 + sin(x))(1 - sin(x))

• (1 - sin(x))(1 + cos(x))

What is the simplified form of the second identity?

• cos(x) - cos^2(x)

• sin(x) + sin^2(x)

• cot(x) - csc(x)

• tan(x) + sec(x)

In the third identity, what is 1 + tan^2(x) equivalent to?

• csc^2(x)

• sec^2(x)

• cot^2(x)

• sin^2(x)