Verifying Trigonometric Identities Using Half Angle Formulas

TL;DR

Learn how to prove various trigonometric identities involving half angle formulas using basic algebraic manipulations.

Transcript

now let's work on a few verifying identities problems associated with half angle formulas so let's prove that sine squared x over 2 is equal to cosecant x minus cotangent x divided by 2 cosecant x now recall that the half angle identity for sine is plus or minus square root 1 minus cosine divided by 2. but here we have sine squared which means we n... Read More

Key Insights

• 👨‍💼 Half angle formulas apply to trigonometric identities involving sine and cosine.
• 🙃 Squaring both sides of a half angle identity can help in proving related identities.
• 😑 Manipulating expressions by multiplying the top and bottom by appropriate factors is a common technique in proving trigonometric identities.

Questions & Answers

Q: How can you prove that sine squared x over 2 is equal to cosecant x minus cotangent x divided by 2 cosecant x?

To prove this identity, square both sides of the half angle identity for sine and manipulate the expression by multiplying the top and bottom by 1 over sine x.

Q: How do you demonstrate that cosine squared x divided by 2 is equal to sine x plus tangent x divided by 2 tan x cosine x over two?

Use the square of the half angle identity for cosine, multiply the top and bottom by sine over cosine to manipulate the expression, and simplify to verify the identity.

Q: What is the best form to use when proving that tangent x divided by two is equal to tangent x divided by secant x plus one?

The form of tangent theta over two as sine theta divided by 1 plus cosine theta is the best to use for this proof because it matches the desired form of the identity.

Q: How can you convert sine x divided by 1 plus cosine x into tangent x divided by secant x plus one?

Multiply the top and bottom of sine x divided by 1 plus cosine x by 1 over cosine x to transform sine into tangent and simplify the expression to obtain tangent x divided by secant x plus one.

Summary & Key Takeaways

• The half angle identity for sine is + or - square root of 1 minus cosine divided by 2.

• To prove sine squared x over 2 is equal to cosecant x minus cotangent x divided by 2 cosecant x, square both sides of the half angle identity for sine.

• To prove cosine squared x divided by 2 is equal to sine x plus tangent x divided by 2 tan x cosine x over two, square both sides of the half angle identity for cosine.