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7.1-7.2 Basic Trigonometric Identities In this powerpoint, we will use trig identities to verify and prove equations. See what you get. Etc. Proving an Identity. Prove the following:. a) sec x (1 + cos x ) = 1 + sec x. = sec x + sec x cos x = sec x + 1. 1 + sec x. - PowerPoint PPT Presentation
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7.1-7.2 Basic Trigonometric Identities
In this powerpoint, we will use trig identities to verify and prove equations
cos
sin1
2 2sin cos 1
1 1 sin cossec csc tan cotcos sin cos sin
1cottan
2 2sin cos 1
2 2
2 2 2
sin cos 1cos cos cos
2 2
2 2 2
sin cos 1sin sin sin
2 2tan 1 sec
2 21 cot csc
1 1 sin cos 1sec csc tan cot cotcos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
sin cot tan sec
cos sinsin sin secsin cos
2sincos sec
cos
2 2cos sin seccos cos
1 sec
cos
sec sec
€
cosθcosθ ⎛ ⎝ ⎜
⎞ ⎠ ⎟•
1 1 sin cos 1sec csc tan cot cotcos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
2 21 cos 1 cot 1 2 2sin csc 1
22
1sin 1sin
1 1
1 1 sin cos 1sec csc tan cot cotcos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
sincsc cot1 cos
1 cos sin
sin sin 1 cos
1 cos1 co
1 cos sinsin 1 coss
21 cos sin
sin 1 cos 1 cos
2sin sin
sin 1 cos 1 cos
sin sin1 cos 1 cos
1 1 sin cos 1sec csc tan cot cotcos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
csc 1 1 sincsc 1 1 sin
sinsin
csc 1 1 sincsc 1 1 sin
1 sin 1 sin1 sin 1 sin
See what you get
1 1 sin cos 1sec csc tan cot cotcos sin cos sin tan
2 2sin cos 1 2 2tan 1 sec 2 21 cot csc
cos 1 sin 2sec1 sin cos
22cos 1 sin2sec
cos 1 sin
2 2cos 1 2sin sin 2sec
cos 1 sin
2 1 sin2sec
cos 1 sin
2 2seccos
2sec 2sec
21 cos csc cot1 cos
21 cos 1 cos csc cot1 cos 1 cos
2
2
2
1 coscsc cot
sin
2
21 cos csc cotsin
2 2csc cot csc cot
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(1
sin x−
cos xsin x
)2
2 2 2
sin cos tancos sin 1 tan
2
2
2 2
2 sin cos ta1
cos1
c
ncos sin 1 tan
os
2 2
2
sintancos
sin 1 tan1cos
2 2
tan tan1 tan 1 tan
22
2
1 cot 2cos 11 cot
22
2
1 cot 2cos 1csc
2
22 2
1 cot 2cos 1csc csc
2 2 2sin cos 2cos 1
2 2sin cos 1 1 1
1 sin 1 sin 4 tan sec1 sin 1 sin
2 2
2
1 sin 1 sin4 tan sec
1 sin
2 2
2
1 2sin sin 1 2sin sin4 tan sec
cos
4sin 4 tan sec
cos cos
4tan sec 4 tan sec
3 3
2
sin cos sec sintan 11 2cos
2 2
2
sin cos sin sin cos cos sec sintan 11 2cos
2
sin cos 1 sin cos sec sintan 11 2cos
2 2 2
sin cos 1 sin cos sec sintan 1sin cos 2cos
2 2
sin cos 1 sin cos sec sintan 1sin cos
sin cos 1 sin cos sec sinsin cos sin cos tan 1
1 1 sin cos sec sincos1 tan 1sin cos
cos
Etc.
5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
=sinxcosx× 1
sinx= 1
cosx=secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x
=sinxcosx×sinx
1 +cosx
=sin2x+cos2xcosx
= 1cosx
=secx
secx
L.S. = R.S.
d) sin4x - cos4x = 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x)= (1 - cos2x - cos2x)= 1 - 2cos2x
L.S. = R.S.
1 - 2cos2x
e)
11+cosx+ 1
1−cosx = 2csc2x
=(1−cosx)+(1+cosx)(1+cosx)(1−cosx)
= 2(1−cos2x)
= 2sin2x
=2csc2x
2csc2x
L.S. = R.S.
Proving an Identity
5.4.9
Proving an Identity
5.4.10
f)
cosA1+sinA+1+sinA
cosA = 2secA
=cos2A+(1+sinA)(1+sinA)(1+sinA)(cosA)
=cos2A+(1+2sinA+sin2A)(1+sinA)(cosA)
=cos2A+sin2A+1+2sinA(1+sinA)(cosA)
= 2+2sinA(1+sinA)(cosA)
= 2(1+sinA)(1+sinA)(cosA)
= 2(cosA)
=2secA
2secA
L.S. = R.S.
