24 Trig Identity Problems

24 Trig Identity Problems
cscθ = 1
sinθ
secθ = 1
cosθ
tanθ = sinθ
cosθ
cot θ = cosθ
sinθ
sin(−θ) = −sinθ
sin(π −θ) = sinθ
sin2 θ + cos2 θ = 1
cos(−θ) = cosθ
cos(π −θ) = −cosθ
tan2 θ + 1= sec2 θ
tan(−θ) = −tanθ
tan(π −θ) = −tanθ
1+ cot 2 θ = csc2 θ
sin(θ +2n π) = sinθ
sin( π2 −θ) = cosθ
cos(θ +2n π) = cosθ
sec( π2 −θ) = cscθ
sin2 θ =
tan(θ +n π) = tanθ
tan( π2 −θ) = cot θ
sin(2θ) = 2sinθ cosθ
cos(2θ) = cos2 θ −sin2 θ
= 2cos2 θ − 1
= 1−2sin2 θ
tan(2θ) = 2 tanθ2
1− tan θ
()
cos( θ ) = ± 1+ cosθ
2
2
tan( θ ) = 1−cosθ
2
sinθ
sin θ = ± 1−cosθ
2
2
=
1−cos(2θ)
2
1+ cos(2θ)
cos2 θ =
2
1−sin2 θ = (1−sinθ)(1+ sinθ) = cos2 θ
1−cos2 θ = (1−cosθ)(1+ cosθ) = sin2 θ
sinαcos β = 21 (sin(α − β)+ sin(α + β))
sinαsinβ = 21 (cos(α − β)−cos(α + β))
cosαcos β = 21 (cos(α − β)+ cos(α + β))
sinθ
1+ cosθ
sin(α + β) = sinαcos β + cosαsinβ
sinα + sinβ = 2sin(α+2 β )cos(α −β
2 )
sin(α − β) = sinαcos β −cosαsinβ
α+ β
sinα −sinβ = 2sin(α −β
2 )cos ( 2 )
cos(α + β) = cosαcos β −sinαsinβ
cos(α − β) = cosαcos β + sinαsinβ
tanα + tanβ
tan(α + β) =
1− tanα tanβ
tanα − tanβ
tan(α − β) =
1+ tanα tanβ
cosα + cos β = 2cos(α+2 β )cos(α −β
2 )
cosα −cos β = −2sin(α+2 β )sin(α −β
2 )
a sinθ + b cosθ = k sin(θ + φ), where k = a 2 + b 2
a
b
sinφ
=
and φ satisfies cosφ =
and
a2 + b 2
a2 + b 2
@blackpenredpen
9/4/2023
1
(Q1.) sin x + cot x cos x =
(A) csc x
(B) sec x
(C) cos x
sec x −cos x
=
sin x
(A) sec x
(B) tan x
(C) cot x
cot x
=
csc x −sin x
(A) tan x
(B) csc x
(C) sec x
1+ 2cos x
=
2 + sec x
(A) cos x
(B) sin x
(C) tan x
1
1
+
=
1−sin x 1+ sin x
(A) 2 tan2 x
(B) 2sec x
(C) 2csc2 x
(B) tan2 x
(C) sin2 x
(Q2.)
(Q3.)
(Q4.)
(Q5.)
2 + cot 2 x
(Q6.)
− 1=
csc2 x
(A) cos2 x
(
2
)
(Q7.) tan x + π =
4
1+ tan x
(A)
1− tan x
(B)
1+ 2 tan x
1− 2 tan x
(C)
1− tan x
1+ tan x
(Q8.) cos(3 x ) =
(A) 2cos3 x + 3cos x
(B) 4 cos3 x −3cos x
(C) −4 cos3 x + cos x
(Q9.) cos(4 x ) =
(A) 4 cos 4 x + 8cos2 x + 1
(B) 8cos 4 x − 4 cos2 x + 3
(C) 8cos 4 x −8cos2 x + 1
(Q10.) sec(sin−1 x ) =
1
(A)
1− x 2
(Q11.) cos(2 tan−1 x ) =
x2
(A)
x2 +1
(B)
x
1− x 2
(C)
1− x 2
(B) 2
x +1
(C)
x
1+ x 2
1−2 x 2
x2 +1
(Q12.) tan(2sin−1 x ) =
(A)
x 1− x 2
1−2 x 2
(B)
2 x 1−2 x 2
1− x 2
2
(C)
2 x 1− x 2
1−2 x 2
(Q13.) csc2 x + sec2x =
(A) csc2 x sec2 x
(Q14.) cos x cos(2 x )
1
(A) (cos x −cos(3 x ))
2
(C) tan2 x
(B) 1
(B)
1
(cos x + cos(3x ))
2
(B)
3
3 1
1
−cos(2 x )−cos(4 x ) (C) − cos(2 x )+ cos(4 x )
4
8 2
8
(B)
−3
1
cos(4 x )−
4
4
(C)
1
3
cos(4 x )+
4
4
(A) tan( x + y )
(B)
sin( x + y )
cos x cos y
(C)
sin( x + y )
cos x + cos y
2 tan x
=
1+ tan2 x
(A) tan(2 x )
(B) cos(2 x )
(C) sin(2 x )
1
1
+
=
sec x − 1 sec x + 1
(A) 2cot x csc x
(B) 2sin x tan x
(C) 2sec2 x
sin x + tan x
=
1+ cos(−x )
(A) sin x
(B) tan x
(C) sin(2 x )
(Q21.) cos2 x −sin4 x sec2 x =
(A) sec2 x
(B) tan2 x
(C) 1− tan2 x
(Q22.) (sin x + cos x )2 =
(A) 1+ sin(2 x )
(B) 1+ cos(2 x )
(C) sin(2 x )+ cos(2 x )
(B) sin x + cos x
(C) sin x + tan x
⎛x ⎞
(B) csc2 ⎜⎜ ⎟⎟
⎝2⎠
(C)
(Q15.) sin4 x =
3
(A) 1− cos(4 x )
8
(Q16.) sin4 x −cos 4 x =
(A) −cos(2 x )
(Q17.) tan x + tan y =
(C) cos(3 x )
(Q18.)
(Q19.)
(Q20.)
sin x
cos x
+
=
1−cot x 1− tan x
(A) sin x −cos x
(Q23.)
⎛x ⎞
(Q24.) sec2 ⎜⎜ ⎟⎟ =
⎝2⎠
cos x
(A)
1−cos x
3
2
1+ cos x
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