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7.1-7.2 Basic Trigonomet ric Identities In this powerpoint, we will use trig identities to verify and prove

# 7.1-7.2 Basic Trigonometric Identities

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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

### Text of 7.1-7.2 Basic Trigonometric Identities

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

(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.

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