Simplifying and Proving a Trigonometric Identity Involving Sine, Cosine, and Tangent Functions

In this article, we will explore the process of simplifying and proving a specific trigonometric identity that involves sine, cosine, and tangent functions. We will break down the steps, use trigonometric identities, and simplify expressions to help you understand the underlying concept thoroughly.

Understanding the Given Identity

The given identity is:

8 ? b ^2 - c ^2 ^2 a 4

This expression is given in the form:

8 b 4 b 4 - c 4 a 4

Using Trigonometric Identities

To simplify the expression, we will use several trigonometric identities to express a, b, and c in terms of simpler functions:

Simplifying a

a can be expressed as:

a sinAsinB 2sin AB 2 cos AB 2

Simplifying b

b can be expressed as:

b tanAtanB sinA sinB cosA cosB

Simplifying c

c can be expressed as:

c cosAcosB cosAcosB 2cos AB 2 cos AB 2 cosAcosB

For simplification, we introduce shorthand notations:

d cosAcosB e AB f A-B

Proving the Identity

Using the shorthand notations, we can write the left-hand side (LHS) as:

LHS 8 . b ^2 - c ^2 ^2 a 4 8 b ^4 - c ^4 a 4 16 sine cos e 2 cos f 2 d 2

Now, let’s simplify the right-hand side (RHS) of the given identity:

RHS 4 b 2 b ^2 - c ^2 ^2 2 4 cos4 e 2 cose cosf 2 d 2 sin2 e d 4 4 4 cos4 e 2 sin2 e d 2

To verify that the LHS equals the RHS, we must show:

2 sin e 2 cos f 2 ? 16 cos e 2 cos f 2 sine d 2 8 b c

which is true even if d, e, and f are arbitrary. Therefore, the given identity holds true.

Conclusion

In this article, we explored the process of simplifying and proving a trigonometric identity that involves sine, cosine, and tangent functions. We used trigonometric identities and introduced shorthand notations to make the expressions easier to handle. We verified the identity by comparing the left-hand side and the right-hand side. This process demonstrates the power of trigonometric identities in simplifying complex expressions.