Question

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

sin(2h)sec(h)+2sin(−h)

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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

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.

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.We will begin with the Pythagorean identities, which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will also use additional identities.