Introduction to Trigonometric Identities - Video Tutorials & Practice Problems
Even and Odd Identities
Example 1
Use the even-odd identities to evaluate the expression.
cos(−θ)−cosθ
0
−cosθ
2cosθ
−2cosθ
Use the even-odd identities to evaluate the expression.
−cot(θ)⋅sin(−θ)
tanθ
−cosθ
cosθ
sin2θcosθ
Select the expression with the same value as the given expression.
sec(−54π)
cos(54π)
−cos(54π)
sec(54π)
−sec(54π)
Select the expression with the same value as the given expression.
sin(−38°)
sin38°
−sin38°
−sin(−38°)
−sin38°1
Pythagorean Identities
Example 2
Use the Pythagorean identities to rewrite the expression as a single term.
(1+cscθ)(1−cscθ)
1
−csc2θ
cot2θ
−cot2θ
Use the Pythagorean identities to rewrite the expression with no fraction.
1−secθ1
1+secθ
tan2θ1
−cot2θ(1+secθ)
−tan2θ(1+secθ)
Simplifying Trig Expressions
Example 3
Example 4
Simplify the expression.
tan2θ−sec2θ+1
0
1
csc2θ+1
2
Simplify the expression.
sec(−θ)tan(−θ)
sinθ
−sinθ
−cotθ
1
Simplify the expression.
(sin2θtan2θ−1)csc2(θ)cos2(−θ)
cot2θ
tanθ
1
– 1
Verifying Trig Equations as Identities
Example 5
Example 6
Identify the most helpful first step in verifying the identity.
(sin2θtan2θ−1)=sec2θsin2(−θ)
Add the terms on the left side using a common denominator.
Rewrite left side of equation in terms of sine and cosine.
Use even-odd identity to eliminate negative argument on right side of equation.
Rewrite right side of equation in terms of sine and cosine.
Identify the most helpful first step in verifying the identity.
sec3θ=secθ+cosθtan2θ
Rewrite left side of equation in terms of sine and cosine.
Subtract secθ from both sides.
Use reciprocal identity to rewrite secθ on right side of equation.
Rewrite tan2θ in terms of sine and cosine.
Do you want more practice?
- In Exercises 1–60, verify each identity. sin x sec x = tan x
- In Exercises 1–60, verify each identity. tan (-x) cos x = -sin x
- In Exercises 1–60, verify each identity. tan x csc x cos x = 1
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Perform each indicated operation and simplify the result so that there are no quotients.sec x/csc x + csc x/se...
- Work each problem.Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, ...
- Perform each indicated operation and simplify the result so that there are no quotients.cos β(sec β + csc β)
- Work each problem.Find the exact values of sin x, cos x, and tan x, for x = π/12 , usinga. difference identiti...
- Perform each indicated operation and simplify the result so that there are no quotients.cos x/sec x + sin x/cs...
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Perform each indicated operation and simplify the result so that there are no quotients.(tan x + cot x)²
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- For each expression in Column I, choose the expression from Column II that completes an identity.2. csc x = __...
- Perform each indicated operation and simplify the result so that there are no quotients.(1 + tan θ)² - 2 tan θ
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Perform each indicated operation and simplify the result so that there are no quotients.1/( sin α - 1) - 1/(si...
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Factor each trigonometric expression.sec² θ - 1
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Factor each trigonometric expression.(tan x + cot x)² - (tan x - cot x)²
- Factor each trigonometric expression.4 tan² β + tan β - 3
- Factor each trigonometric expression.cot⁴ x + 3 cot² x + 2
- Factor each trigonometric expression.sin³ α + cos³ α
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- For each expression in Column I, choose the expression from Column II that completes an identity.4. cot x = __...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- Concept Check Suppose that sec θ = (x+4)/x.Find an expression in x for tan θ.
- Verify that each equation is an identity.tan α/sec α = sin α
- Perform each transformation. See Example 2.Write cot x in terms of sin x.
- Verify that each equation is an identity.(tan² α + 1)/ sec α = sec α
- Perform each transformation. See Example 2.Write cot x in terms of csc x.
- Verify that each equation is an identity.sin² β (1 + cot² β) = 1
- Verify that each equation is an identity.2 cos³ x - cos x = (cos² x - sin² x)/sec x
- Perform each transformation. See Example 2.Write sec x in terms of sin x.
- Verify that each equation is an identity.sin² α + tan² α + cos² α = sec² α
- Verify that each equation is an identity.(sin 2x)/(sin x) = 2/sec x
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(sin² θ)/cos θ = sec θ - cos θ
- Verify that each equation is an identity.(2 tan B)/(sin 2B) = sec² B
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.sec⁴ x - sec² x = tan⁴ x + tan² x
- Verify that each equation is an identity.(2 cot x)/(tan 2x) = csc² x - 2
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(sec α - tan α)² = (1 - sin α)/(1 + sin α)
- Verify that each equation is an identity.csc A sin 2A - sec A = cos 2A sec A
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- For each expression in Column I, choose the expression from Column II that completes an identity.6. sec² x = _...
- Verify that each equation is an identity.[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
- Verify that each equation is an identity.2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.1/(sec α - tan α) = sec α + tan α
- Verify that each equation is an identity.sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ
- Verify that each equation is an identity.sin³ θ = sin θ - cos² θ sin θ
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.sin² θ (1 + cot² θ) - 1 = 0
- Verify that each equation is an identity.2 cos² (x/2) tan x = tan x+ sin x
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
- Verify that each equation is an identity.(1/2)cot (x/2) - (1/2) tan (x/2) = cot x
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
- Verify that each equation is an identity.(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.tan² α sin² α = tan² α + cos² α - 1
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ) ...
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)
- Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Verify that each equation is an identity.(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
- Verify that each equation is an identity.(sec α + csc α) (cos α - sin α) = cot α - tan α
- Verify that each equation is an identity.(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1
- Verify that each equation is an identity.sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x
- Verify that each equation is an identity.sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)
- In Exercises 1–60, verify each identity. csc θ - sin θ = cot θ cos θ
- In Exercises 1–60, verify each identity. cos θ sec θ ----------------- = tan θ cot θ
- In Exercises 1–60, verify each identity. cos² θ (1 + tan² θ) = 1
- In Exercises 1–60, verify each identity. cot² t ------------ = csc t - sin t csc t