Verify which of the following are identities

Now our numerator is equal to sine squared that we know now. I'm glad I didn't multiply that denominator out because I am able to cancel one sign data, so we'll get rid of that squared. Get rid of that sign. Data on the bottom. We can divide those and get a one. So now we simply have science data over one minus co. Sign of data. Which is exactly what. Click 'Join' if it's correct. Laura J. Algebra and Trigonometry 2 months ago. View Full Video Already have an account? Ashley C. Answer Verify that the following equations are identities.

Section 2 Constructing and Verifying Identities. Discussion You must be signed in to discuss. Video Transcript okay for this question. Upgrade today to get a personal Numerade Expert Educator answer! Ask unlimited questions. Test yourself. Join Study Groups. Create your own study plan. Join live cram sessions. Live student success coach. University of North Carolina at Chapel Hill.

Kayleah T. Harvey Mudd College. Kristen K. University of Michigan - Ann Arbor. Michael D. Utica College. Recommended Videos The other even-odd identities follow from the even and odd nature of the sine and cosine functions. The cosecant function is therefore odd. The secant function is therefore even.

To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.

The next set of fundamental identities is the set of reciprocal identities , which, as their name implies, relate trigonometric functions that are reciprocals of each other. The final set of identities is the set of quotient identities , which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle.

The quotient identities define the relationship among the trigonometric functions. We see only one graph because both expressions generate the same image. One is on top of the other.

This is a good way to prove any identity. If both expressions give the same graph, then they must be identities. There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side. In the second method, we split the fraction, putting both terms in the numerator over the common denominator.

This problem illustrates that there are multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer will be the same. There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression:.

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations. We can also create our own identities by continually expanding an expression and making the appropriate substitutions.

Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve. Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1.

This is the difference of squares. If this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We have. All of the Pythagorean identities are related. For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

For the following exercises, determine whether the identity is true or false.