Wednesday, September 9, 2009

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

The mathematics question is, prove the following identities:


Additional Mathematics
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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

= RHS (answer)



(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

= RHS (answer)




(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

= RHS (answer)





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





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Reference

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