Solved: Trigonometric Substitution Make the indicatedtrigonometric substitution in the

An equation is called an identity if it is valid for values of the variable. The equation 2x x x is an algebraic identity, and the equation sin2 x cos2 x is a trigonometric identity.

For any x it is true that cos1x2 has the same value as cos x. We express this fact as the identity

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. cos t tan t

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. cos t csc t

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. sin u sec u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. tan u csc u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. tan u csc u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. tan u csc u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. sin u cot u cos u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. cos2 u 11 tan2 u2

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. sec u cos u sin u

Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. cot u csc u sin u

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin x sec x tan x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin x sec x tan x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin t tan t tan t

Simplifying Trigonometric Expressions Simplify the trigonometric expression. 1 cot A csc A

Simplifying Trigonometric Expressions Simplify the trigonometric expression. cos3x sin2 x cos x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin4 a cos4 a cos2 a

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sec2 x 1 sec2 x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sec x cos x tan x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. 1 cos y 1 sec y

Simplifying Trigonometric Expressions Simplify the trigonometric expression. 1 sin y 1 csc y

Simplifying Trigonometric Expressions Simplify the trigonometric expression. 1 sin u cos u cos u 1 sin u

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin t 1 cos t csc t

Simplifying Trigonometric Expressions Simplify the trigonometric expression. cos x sec x tan x

Simplifying Trigonometric Expressions Simplify the trigonometric expression. cot A 1 1 tan1A2

Simplifying Trigonometric Expressions Simplify the trigonometric expression. sin a 1 1 sin a

Simplifying Trigonometric Expressions Simplify the trigonometric expression. 2 tan2 x sec2 x 1

Proving an Identity Algebraically and Graphically Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. cos x sec x sin x csc x sin x

Proving an Identity Algebraically and Graphically Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. tan y csc y sec y cos y

Proving Identities Verify the identity. sin u tan u cos u

Proving Identities Verify the identity. tan x sec x sin x

Proving Identities Verify the identity. cos u sec u tan u cot u

Proving Identities Verify the identity. cot x sec x csc x 1

Proving Identities Verify the identity. tan y csc y 1 cos y 1 sec y

Proving Identities Verify the identity. cos2 sin csc sin

Proving Identities Verify the identity. cos1x2 sin1x2 cos x sin x

Proving Identities Verify the identity. cot1a2 cos1a2 sin1a2 csc a

Proving Identities Verify the identity.tan u cot u sec u csc u

Proving Identities Verify the identity. 1sin x cos x22 1 2 sin x cos x

Proving Identities Verify the identity. 11 cos b2 11 cos b2 1 csc2 b

Proving Identities Verify the identity. cos x sec x sin x csc x 1

Proving Identities Verify the identity. 1 1 sin2 y 1 tan2 y

Proving Identities Verify the identity.csc x sin x cos x cot x

Proving Identities Verify the identity. 1tan x cot x22 sec2 x csc2 x

Proving Identities Verify the identity. tan2 x cot2 x sec2 x csc2 x

Proving Identities Verify the identity. 11 sin2 t cos2 t 22 4 sin2 t cos2 t 4 cos2 t

Proving Identities Verify the identity. 2 sin x cos x 1sin x cos x22 1 1

Proving Identities Verify the identity. csc x cos2 x sin x csc x

Proving Identities Verify the identity. cot2 t cos2 t cot2 t cos2 t

Proving Identities Verify the identity. 1sin x cos x22 sin2 x cos2 x sin2 x cos2 x 1sin x cos x22

Proving Identities Verify the identity. 1sin x cos x24 11 2 sin x cos x22

Proving Identities Verify the identity. sec t cos t sec t sin2 t

Proving Identities Verify the identity. 1cot x csc x2 1cos x 12 sin x

Proving Identities Verify the identity. cos2 x sin2 x 2 cos2 x 1

Proving Identities Verify the identity. 2 cos2 x 1 1 2 sin2 x

Proving Identities Verify the identity. sin4 u cos4 u sin2 u cos2 u

Proving Identities Verify the identity. 11 cos2 x2 11 cot2 x2 1

Proving Identities Verify the identity. 1sin t cos t 22 sin t cos t 2 sec t csc t

Proving Identities Verify the identity. sec t csc t 1tan t cot t 2 sec2 t csc2 t

Proving Identities Verify the identity. 1 tan2 u 1 tan2 u 1 cos2 u sin2 u

Proving Identities Verify the identity. 1 sec2 x 1 tan2 x 1 cos2 x

Proving Identities Verify the identity.sec x csc x tan x cot x sin x cos x

Proving Identities Verify the identity. sin x cos x sec x csc x sin x cos x

Proving Identities Verify the identity.1 cos x sin x sin x 1 cos x 2 csc x

Proving Identities Verify the identity. csc x cot x sec x 1 cot x

Proving Identities Verify the identity. tan2 u sin2 u tan2 u sin2 u

Proving Identities Verify the identity. sec4 x tan4 x sec2 x tan2 x

Proving Identities Verify the identity. 1 tan x 1 tan x cos x sin x cos x sin x

Proving Identities Verify the identity. cos u 1 sin u sin u csc u cos u cot u

Proving Identities Verify the identity. 1 sec x tan x 1 sec x tan x 2 sec x

Proving Identities Verify the identity.cos2 t tan2 t 1 sin2 t tan2 t

Proving Identities Verify the identity. 1 sin x 1 sin x 1 sin x 1 sin x 4 tan x sec x

Proving Identities Verify the identity.tan x tan y cot x cot y tan x tan y

Proving Identities Verify the identity. sin3 x cos3 x sin x cos x 1 sin x cos x

Proving Identities Verify the identity. tan cot tan2 cot2 sin cos

Proving Identities Verify the identity. 1 cos a sin a sin a 1 cos a

Proving Identities Verify the identity. sin x 1 sin x 1 cos2 x 1sin x 122

Proving Identities Verify the identity. sin sin cos tan 1 tan

Proving Identities Verify the identity. sin A 1 cos A cot A csc A

Proving Identities Verify the identity. sec x sec x tan x sec x 1sec x tan x2

Proving Identities Verify the identity. sec tan 1 sec tan

Proving Identities Verify the identity. cos u 1 sin u sec u tan u

Proving Identities Verify the identity. tan sin tan sin tan sin tan sin

Proving Identities Verify the identity. 1 sin x 1 sin x 1sec x tan x22

Proving Identities Verify the identity. 1 sin x 1 sin x 1tan x sec x22

Proving Identities Verify the identity. csc x cot x 1 csc x cot x

Proving Identities Verify the identity. sec u 1 sec u 1 tan u sin u tan u sin u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. x "1 x 2 , x sin u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. "1 x 2, x tan u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. "x 2 1, x sec u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. 1 x 2"4 x 2 , x 2 tan u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. "9 x 2, x 3 sin u

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 , u p/2. "x 2 25 x , x 5 sec u

Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f1x2 g1x2 is an identity? Prove your answer. f1x2 cos2 x sin2 x, g1x2 1 2 sin2 x

Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f1x2 g1x2 is an identity? Prove your answer. f1x2 tan x 11 sin x2, g1x2 sin x cos x 1 sin x

Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f1x2 g1x2 is an identity? Prove your answer. f1x2 1sin x cos x22, g1x2 1

Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f1x2 g1x2 is an identity? Prove your answer. f1x2 1sin x cos x22, g1x2 1

Proving More Identities Verify the identity. 1sin x sin y cos x cos y2 1sin x sin y cos x cos y2 sin2 y cos2 x

Proving More Identities Verify the identity. 1 cos x sin x 1 cos x sin x 1 sin x cos x

Proving More Identities Verify the identity. 1tan x cot x24 sec4 x csc4 x

Proving More Identities Verify the identity. 1sin a tan a2 1cos a cot a2 1cos a 12 1sin a 12

Proving More Identities Verify the identity. 1sin a tan a2 1cos a cot a2 1cos a 12 1sin a 12

Proving More Identities Verify the identity. sin6 b cos6 b 1 3 sin2 b cos2 b

Proving Identities Involving Other Functions These identities involve trigonometric functions as well as other functions that we have studied. Proving Identities Involving Other Functions These identities involve trigonometric functions as well as other functions that we have studied.

Proving Identities Involving Other Functions These identities involve trigonometric functions as well as other functions that we have studied. ln 0 tan x 0 ln 0 cot x 0 0

Proving Identities Involving Other Functions These identities involve trigonometric functions as well as other functions that we have studied. esin2 xetan2 x esec2 xecos2 x

Proving Identities Involving Other Functions These identities involve trigonometric functions as well as other functions that we have studied. ex2 ln 0 sin x0 ex sin2 x

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions. esin2 xecos2 x e

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions. x x 1 1 x

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions. "sin2 x 1 "sin2 x 1

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions. xeln x2 x3

An Identity Involving Three Variables Suppose x R cos u sin f, y R sin u sin f, and z R cos f. Verify the identity x2 y2 z2 R2.

Discuss: Equations That Are Identities You have encountered many identities in this course. Which of the following equations do you recognize as identities? For those that you think are identities, test several values of the variables to confirm that the equation is true for those variables. (a) 1x y22 x2 2xy y2 (b) x2 y2 1 (c) x1y z2 xy xz (d) t2 cos2 t 1t cos t 2 1t cos t 2 (e) sin t cos t 1 (f) x2 tan2 x 0

Discuss: Equations That Are Not Identities How can you tell if an equation is not an identity? Show that the following equations are not identities. (a) sin 2x 2 sin x (b) sin1x y2 sin x sin y (c) sec2 x csc2 x 1 (d) 1 sin x cos x csc x sec x

Discuss: Graphs and Identities Suppose you graph two functions, f and g, on a graphing device and their graphs appear identical in the viewing rectangle. Does this prove that the equation f1x2 g1x2 is an identity? Explain.

DiscOV ER: Making Up Your Own Identity If you start with a trigonometric expression and rewrite it or simplify it, then setting the original expression equal to the rewritten expression yields a trigonometric identity. For instance, from Example 1 we get the identity cos t tan t sin t sec t Use this technique to make up your own identity, then give it to a classmate to verify.

Discuss: Cofunction Identities In the right triangle shown, explain why 1p/22 u. Explain how you can obtain all six cofunction identities from this triangle for 0 u p/2. Note that u and are complementary angles. So the cofunction identities state that a trigonometric function of an angle u is equal to the corresponding cofunction of the complementary angle .