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 = ———
                                                       sin²Ө

Solution:

LHS = (1 + cotӨ)² + (1 - cotӨ )²

= 1 + 2cotӨ + cot²Ө + 1 - 2cotӨ + cot²Ө

= 2 + 2cot²Ө

               cos²Ө
= 2 + 2( ———— )
                sin²Ө

  2sin²Ө + 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

= tan²Ө 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

• 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 - 183
• http://www.singaporemath.com/PhotoGallery.asp?ProductCode=NSAMT
• http://www.intmath.com/Analytic-trigonometry/1_Trigonometric-identities.php
• http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities
• http://www.mathematics.com.au/images/600x480/5558.gif
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