Question

What is the result when the following expression is simplified as much as possible?

sin(2h)sec(h)+2sin(−h)​

Answer 1

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Because sin x is an odd function, we can rewrite the second term in the expression.

2sin(−h)=−2sin h.

We now use a double-angle formula to expand the first term.

sin(2h)sec h=2sin h cos h sec h.

Because they are reciprocals, cos h sec h=1.

2sin h cos h sec h−2sin h=2sin h−2sin h=0.

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Answer 2

Answer:

Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways.

To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. In this first section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities.