3.1. Fundamental Identities www.ck12.org

3.1 Fundamental Identities

1. tan270◦ = sin270◦cos270◦ =

−10 , you cannot divide by zero, therefore tan270◦ is undefined.

2. If cos(

π

2 − x)= 4

5 , then, by the cofunction identities, sinx = 45 . Because sine is odd, sin(−x) =−4

5 .3. If tan(−x) =− 5

12 , then tanx = 512 . Because sinx =− 5

13 , cosine is also negative, so cosx =−1213 .

4. Use the reciprocal and cofunction identities to simplify

secxcos(

π

2− x)

1cosx

· sinx

sinxcosxtanx

5. (a) Using the sides 5, 12, and 13 and in the first quadrant, it doesn’t really matter which is cosine or sine.

• So, sin2θ+ cos2 θ = 1 becomes

( 513

)2+(12

13

)2= 1. *Simplifying, we get: 25

169 +144169 = 1,

• Finally 169169 = 1.

(b) sin2θ+ cos2 θ = 1 becomes

(12

)2+

(√3

2

)2

= 1. Simplifying we get: 14 +

34 = 1 and 4

4 = 1.

6. To prove tan2 θ+1 = sec2 θ, first use sinθ

cosθ= tanθ and change sec2 θ = 1

cos2 θ.

tan2θ+1 = sec2

θ

sin2θ

cos2 θ+1 =

1cos2 θ

sin2θ

cos2 θ+

cos2 θ

cos2 θ=

1cos2 θ

sin2θ+ cos2

θ = 1

7. If cscz = 178 and cosz =−15

17 , then sinz = 817 and tanz =− 8

15 . Therefore cotz =−158 .

8. (a) Factor sin2θ− cos2 θ using the difference of squares.

sin2θ− cos2

θ = (sinθ+ cosθ)(sinθ− cosθ)

(b) sin2θ+6 sinθ+8 = (sinθ+4)(sinθ+2)

9. You will need to factor and use the sin2θ+ cos2 θ = 1 identity.

sin4θ− cos4 θ

sin2θ− cos2 θ

=(sin2

θ− cos2 θ)(sin2θ+ cos2 θ)

sin2θ− cos2 θ

= sin2θ+ cos2

θ

= 1

10. To rewrite cosxsecx−1 so it is only in terms of cosine, start with changing secant to cosine.

cosxsecx−1

=cosx1

cosx −1Now, simplify the denominator.

cosx1

cosx −1=

cosx1−cosx

cosx

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www.ck12.org Chapter 3. Trigonometric Identities and Equations, Solution Key

Multiply by the reciprocal cosx1−cosx

cosx= cosx÷ 1−cosx

cosx = cosx · cosx1−cosx =

cos2 x1−cosx

11. The easiest way to prove that tangent is odd to break it down, using the Quotient Identity.

tan(−x) =sin(−x)cos(−x)

from this statement, we need to show that tan(−x) =− tanx

=−sinxcosx

because sin(−x) =−sinx and cos(−x) = cosx

=− tanx

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