## Wednesday, September 9, 2009

### Mathematics - Trigonometrical Ratios and Equations (6.11 - Proving of Identities)

The mathematics question is, prove the following identities:

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2
(n) (1 + cotӨ)
2 + (1 - cotӨ )2 = ———
s
in
2Ө

Solution:

LHS =
(1 + cotӨ)2 + (1 - cotӨ )2

=
1 + 2cot
Ө + cot2Ө + 1 - 2cotӨ + cot2Ө

= 2 + 2
cot2Ө

cos2Ө
= 2 + 2(
——— )
s
in
2Ө

2
sin2Ө + 2cos2Ө
=
————————

sin2Ө

2(sin2Ө + cos2Ө)
= ————————

sin2Ө

2
= ———

sin2Ө

(o) tan2Ө - sin2Ө = tan2Ө sin2Ө

Solution:
LHS = tan2Ө - sin2Ө

sin2Ө
= ———  sin2Ө

cos2Ө

sin2Ө - sin2Ө cos2Ө
= ————————
cos2Ө

sin2Ө (1 - cos2Ө)
= ————————
cos2Ө

sin2Ө sin2Ө
= ——————
cos2Ө

= tan
2Ө sin2Ө

(q) (1 - cosx)(1 + secx) = sinx tanx

Solution:
LHS = (1 - cosx)(1 + secx)

1
=
(1 - cosx)( 1 + )
cosx

1
= (1 - cosx) + (1 - cosx)
cosx

1
= 1 - cosx +   - 1
cosx

cosx - cos2x + 1 - cosx
= ————————
cosx

1 - cos2x
= —————

cosx

sin2x
= ———
cosx

sinx
= sinx ——
cosx

= sinx tanx

Standard Trigonometrical Ratios and Identities

sinA                                             1
tanA = ——                     cosecA = ——                  sin2A + cos2A = 1
cosA                                         sinA

cosA
1
cotA =
———                     secA =
——                       1 + tan2A = sec2A
sinA
cosA

1
cotA = ——                        1 + cot2A = cosec2A

tanA

From "Additional Mathematics", 8th Edition, by The Keng Seng, Loh Cheng Yee, Consultant: Dr. Yeap Ban Har, Shinglee Publishers Pte Ltd, Chapter 6 - “Trigonometrical Ratios and Equations”, 6.11 - Proving of Identities, Page 180, table 6.1

Obtained from
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Obtained from http://www.trigonometry-help.net/img/trig-tables_clip_image018.gif

Reference