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Determine which of the following are trigonometric identities. I. cos⁡(t) −cos⁡(s) sin⁡(t) +sin⁡(s) +sin⁡(t) −sin⁡(s) cos⁡(t) +cos⁡(s) =0\frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 0 II. cos⁡(t) +cos⁡(s) sin⁡(t) +sin⁡(s) +sin⁡(t) +sin⁡(s) cos⁡(t) +cos⁡(s) =1\frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 1 III. cos⁡(t) +sin⁡(s) cos⁡(t) sin⁡(s) =cos⁡(s) +sin⁡(t) \frac { \cos ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) \sin ( \mathrm { s } ) } = \cos ( \mathrm { s } ) + \sin ( \mathrm { t } )


A) I is the only identity.
B) I, II, and III are identities.
C) II is the only identity.
D) II and II are the only identities.
E) I and II are the only identities.

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Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. sin⁡α(csc⁡α−sin⁡α) \sin \alpha ( \csc \alpha - \sin \alpha )


A) 1−sin⁡2α1 - \sin ^ { 2 } \alpha
B) csc⁡2α−1csc⁡2α\frac { \csc ^ { 2 } \alpha - 1 } { \csc ^ { 2 } \alpha }
C) csc⁡2α−sec⁡2α+tan⁡2αcsc⁡2α\frac { \csc ^ { 2 } \alpha - \sec ^ { 2 } \alpha + \tan ^ { 2 } \alpha } { \csc ^ { 2 } \alpha }
D) 1−cot⁡2α1 - \cot ^ { 2 } \alpha
E) cos⁡2α\cos ^ { 2 } \alpha

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Verify the given identity. cos⁡u−cos⁡vcos⁡u+cos⁡v=−tan⁡12(u+v)tan⁡12(u−v)\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )

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\[\begin{aligned} \frac { \cos u - \cos v } { \cos u + \cos v } & = \frac { - 2 \sin \left( \frac { u + v } { 2 } \right) \sin \left( \frac { u - v } { 2 } \right) } { 2 \cos \left( \frac { u + v } { 2 } \right) \cos \left( \frac { u - v } { 2 } \right) } \\ & = - \tan \left( \frac { u + v } { 2 } \right) \tan \left( \frac { u - v } { 2 } \right) \\ & = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v ) \end{aligned}\]

Use the product-to-sum formula to write the given product as a sum or difference. 8sin⁡π8sin⁡π88 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }


A) 4sin⁡π164 \sin \frac { \pi } { 16 }
B) 4−4cos⁡π44 - 4 \cos \frac { \pi } { 4 }
C) 4+4cos⁡π164 + 4 \cos \frac { \pi } { 16 }
D) −4sin⁡π16- 4 \sin \frac { \pi } { 16 }
E) 4sin⁡π8+4cos⁡π84 \sin \frac { \pi } { 8 } + 4 \cos \frac { \pi } { 8 }

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Use the half-angle formulas to determine the exact value of the following. cos⁡(22.5∘) \cos \left( 22.5 ^ { \circ } \right)


A) −2+32- \frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }
B) 2−22\frac { \sqrt { 2 - \sqrt { 2 } } } { 2 }
C) −2−22- \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 }
D) 3−32\frac { \sqrt { 3 - \sqrt { 3 } } } { 2 }
E) 2+22\frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }

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Verify the identity shown below. tan⁡θ+1sec⁡θ+csc⁡θ=sin⁡θ\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta

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\[\begin{aligned}
\frac { \tan \theta + ...

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Find the exact value of tan⁡(u+v) \tan ( u + v ) given that sin⁡u=−1161\sin u = - \frac { 11 } { 61 } and cos⁡v=4041\cos v = \frac { 40 } { 41 } . (Both uu and vv are in Quadrant IV.)


A) tan⁡(u+v) =37767\tan ( u + v ) = \frac { 37 } { 767 }
B) tan⁡(u+v) =24882301\tan ( u + v ) = \frac { 2488 } { 2301 }
C) tan⁡(u+v) =−9802301\tan ( u + v ) = - \frac { 980 } { 2301 }
D) tan⁡(u+v) =797767\tan ( u + v ) = \frac { 797 } { 767 }
E) tan⁡(u+v) =−833767\tan ( u + v ) = - \frac { 833 } { 767 }

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Expand the expression below and use fundamental trigonometric identities to simplify. (sin⁡(ω) +cos⁡(ω) ) 2( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }


A) sin⁡2(ω) +cos⁡2(ω) \sin ^ { 2 } ( \omega ) + \cos ^ { 2 } ( \omega )
B) 2tan⁡(ω) +12 \tan ( \omega ) + 1
C) 2sin⁡(ω) cos⁡(ω) +12 \sin ( \omega ) \cos ( \omega ) + 1
D) 1
E) 2cot⁡(ω) +12 \cot ( \omega ) + 1

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C

If x=10sin⁡θx = 10 \sin \theta , use trigonometric substitution to write 100−x2\sqrt { 100 - x ^ { 2 } } as a trigonometric function of θ\theta , where −π2<θ<π2- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .


A) 10sin⁡θ10 \sin \theta
B) 10cos⁡θ10 \cos \theta
C) 10tan⁡θ10 \tan \theta
D) 10csc⁡θ10 \csc \theta
E) 10sec⁡θ10 \sec \theta

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Use the sum-to-product formulas to write the given expression as a product. cos⁡8θ−cos⁡6θ\cos 8 \theta - \cos 6 \theta


A) −2sin⁡7θsin⁡θ- 2 \sin 7 \theta \sin \theta
B) 2cos⁡7θcos⁡θ2 \cos 7 \theta \cos \theta
C) 2cos⁡7θsin⁡θ2 \cos 7 \theta \sin \theta
D) 2sin⁡7θcos⁡θ2 \sin 7 \theta \cos \theta
E) −2cos⁡7θcos⁡θ- 2 \cos 7 \theta \cos \theta

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Write the given expression as the sine of an angle. sin⁡85∘cos⁡50∘+sin⁡50∘cos⁡85∘\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }


A) sin⁡(−100∘) \sin \left( - 100 ^ { \circ } \right)
B) sin⁡(135∘) \sin \left( 135 ^ { \circ } \right)
C) sin⁡(35∘) \sin \left( 35 ^ { \circ } \right)
D) sin⁡(85∘) \sin \left( 85 ^ { \circ } \right)
E) sin⁡(50∘) \sin \left( 50 ^ { \circ } \right)

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Verify the identity shown below. tan⁡α+cot⁡βtan⁡αcot⁡β=tan⁡β+cot⁡α\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha

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\[\begin{aligned}
\frac { \tan \alpha + ...

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Find the exact solutions of the given equation in the interval [0,2π) [ 0,2 \pi ) .


A) x=0,π3,2π3,π,4π3x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }
B) x=π6,π2,5π6,3π2x = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }
C) x=0,π2,π,3π2x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }
D) x=π6,5π6,3π2x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }
E) x=π4,3π4,5π4,7π4x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }

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Expand the expression below and use fundamental trigonometric identities to simplify. (sin⁡(ω) +cos⁡(ω) ) 2( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }


A) sin⁡2(ω) +cos⁡2(ω) \sin ^ { 2 } ( \omega ) + \cos ^ { 2 } ( \omega )
B) 2tan⁡(ω) +12 \tan ( \omega ) + 1
C) 2sin⁡(ω) cos⁡(ω) +12 \sin ( \omega ) \cos ( \omega ) + 1
D) 1
E) 2cot⁡(ω) +12 \cot ( \omega ) + 1

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Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. (sin⁡x+cos⁡x) (sin⁡x−cos⁡x) ( \sin x + \cos x ) ( \sin x - \cos x )


A)
2sin⁡2x−sec⁡2x−tan⁡2x2 \sin ^ { 2 } x - \sec ^ { 2 } x - \tan ^ { 2 } x
B)
sin⁡2x−cos⁡2x\sin ^ { 2 } x - \cos ^ { 2 } x
C)
1−2cos⁡2x1 - 2 \cos ^ { 2 } x
D)
csc⁡2x−cot⁡2x−2cos⁡2x\csc ^ { 2 } x - \cot ^ { 2 } x - 2 \cos ^ { 2 } x
E)
1−2sin⁡(π2−x) cos⁡x1 - 2 \sin \left( \frac { \pi } { 2 } - x \right) \cos x

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Use the half-angle formulas to determine the exact value of the following. cos⁡(−22.5∘) \cos \left( - 22.5 ^ { \circ } \right)


A) −2+22- \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }
B) 2−32\frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }
C) −2−32- \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }
D) 3−22\frac { \sqrt { 3 - \sqrt { 2 } } } { 2 }
E) 2+32\frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }

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Determine which of the following are trigonometric identities. 6 I. cos⁡(4x) +cos⁡(2x) 2cot⁡(3x) =cot⁡(x) \frac { \cos ( 4 x ) + \cos ( 2 x ) } { 2 \cot ( 3 x ) } = \cot ( x ) II. cos⁡(4x) +cos⁡(x) sin⁡(3x) −sin⁡(x) =cot⁡(2x) \frac { \cos ( 4 x ) + \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = \cot ( 2 x ) III. cos⁡(6x) +cos⁡(2x) sin⁡(4x) +sin⁡(2x) =cot⁡(3x) \frac { \cos ( 6 x ) + \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = \cot ( 3 x )


A) III is the only identity.
B) I, II, and III are identities.
C) I is the only identity.
D) None are identities.
E) II is the only identity.

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A

Find the exact value of tan⁡(u+v) \tan ( u + v ) given that sin⁡u=−35\sin u = - \frac { 3 } { 5 } and cos⁡v=2425\cos v = \frac { 24 } { 25 } . (Both uu and vv are in Quadrant IV.)


A) tan⁡(u+v) =4175\tan ( u + v ) = \frac { 41 } { 75 }
B) tan⁡(u+v) =3875\tan ( u + v ) = \frac { 38 } { 75 }
C) tan⁡(u+v) =−43\tan ( u + v ) = - \frac { 4 } { 3 }
D) tan⁡(u+v) =8975\tan ( u + v ) = \frac { 89 } { 75 }
E) tan⁡(u+v) =−3925\tan ( u + v ) = - \frac { 39 } { 25 }

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Write the given expression as an algebraic expression. cos⁡(arccos⁡x−arcsin⁡x) \cos ( \arccos x - \arcsin x )


A) x1−x2−xx2+1\frac { x \sqrt { 1 - x ^ { 2 } } - x } { \sqrt { x ^ { 2 } + 1 } }
B) 2x1−x22 x \sqrt { 1 - x ^ { 2 } }
C) x1−x2+xx2+1\frac { x \sqrt { 1 - x ^ { 2 } } + x } { \sqrt { x ^ { 2 } + 1 } }

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Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. sin⁡x+1+1sin⁡x−1\sin x + 1 + \frac { 1 } { \sin x - 1 }


A) sin⁡2xsin⁡x−1\frac { \sin ^ { 2 } x } { \sin x - 1 }
B) 1−cos⁡2xsin⁡x−1\frac { 1 - \cos ^ { 2 } x } { \sin x - 1 }
C) 1+cos⁡2xsin⁡x−1\frac { 1 + \cos ^ { 2 } x } { \sin x - 1 }
D) sec⁡x−cos⁡xtan⁡x−sec⁡x\frac { \sec x - \cos x } { \tan x - \sec x }
E) csc⁡x−cot⁡xcos⁡x1−csc⁡x\frac { \csc x - \cot x \cos x } { 1 - \csc x }

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