Rewriting and Simplifying Trigonometric Identities

Introduction

In this article, we will explore how to rewrite and simplify trigonometric identities, specifically focusing on sin and cos functions. We'll provide a detailed explanation of the steps involved in transforming and simplifying given trigonometric expressions.

Simplifying a Given Expression

Let's start with the expression: (y arcsin{4w}).

The identity given is: ( cos{2x} 1 - 2sin^2{x} ).

We can substitute (y arcsin{4w}) into this identity.

Since (y arcsin{4w}), we have:

( sin{y} 4w )

Using the double angle formula for cosine, we have:

( cos{2y} 1 - 2sin^2{y} )

Substitute ( sin{y} 4w ) into the formula:

( cos{2y} 1 - 2(4w)^2 )

( cos{2y} 1 - 32w^2 )

Exploring Another Identity

Let's now consider another expression: ( theta sin^{-1}{4w} ).

We have ( sin{theta} 4w ).

Using the Pythagorean identity, we know that ( cos^2{theta} sin^2{theta} 1 ).

Therefore,

( cos^2{theta} 1 - sin^2{theta} 1 - (4w)^2 1 - 16w^2 )

We recognize ( cos{theta} ) can be positive or negative, as:

( cos{theta} pm sqrt{1 - 16w^2} )

Now, we can use the double angle formula for cosine again:

( cos{2theta} cos^2{theta} - sin^2{theta} )

Substituting the values we have:

( cos{2theta} (1 - 16w^2) - (16w^2) )

( cos{2theta} 1 - 32w^2 )

Conclusion

In this article, we have demonstrated how to rewrite and simplify trigonometric identities involving arcsin and cos2x. We applied the double angle formula and the Pythagorean identity to derive the final expressions.

Understanding these identities is crucial for solving a wide range of trigonometric problems and simplifying complex expressions. Whether you are a student or a professional dealing with trigonometric equations, this knowledge will be invaluable.