J's answer to Nicole She's Secondary 3 A Maths Singapore question.
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Continued from your working :
= (sinx(secx + 1) + sinx(secx - 1))/[(secx + 1)(secx - 1)]
= (sinxsecx + sinx + sinxsecx - sinx)/(sec²x -1²)
= 2sinxsecx/tan²x
= 2sinx(1/cosx) ÷ sin²x/cos²x
= 2sinx/cosx × cos²x/sin²x
= 2cosx/sinx
= 2cotx
= (sinx(secx + 1) + sinx(secx - 1))/[(secx + 1)(secx - 1)]
= (sinxsecx + sinx + sinxsecx - sinx)/(sec²x -1²)
= 2sinxsecx/tan²x
= 2sinx(1/cosx) ÷ sin²x/cos²x
= 2sinx/cosx × cos²x/sin²x
= 2cosx/sinx
= 2cotx
Date Posted:
4 years ago
You have to use the property
1 + tan²x = sec²x
and
(a - b)(a + b) = a² - b²
1 + tan²x = sec²x
and
(a - b)(a + b) = a² - b²
Nicole, in such proving the identities questions, the following techniques are useful.
1. Combining into single fraction
2. Usage of three trigonometric identities
3. Conversion of the terms into expressions involving sines and cosines only
4. Introduction of terms on both numerator and denominator, akin to rationalising a denominator in surds
5. Observing the connection between terms for each side (eg if the left side has a plus sign connecting two terms and the right side does not contain one, but instead contains a product of terms, the priority is to eliminate the plus signs)
6. Usage of basic algebraic-like identities such as a^2 - b^2 = (a + b) (a - b)
There are other techniques too, which you will face when proving such identities.
1. Combining into single fraction
2. Usage of three trigonometric identities
3. Conversion of the terms into expressions involving sines and cosines only
4. Introduction of terms on both numerator and denominator, akin to rationalising a denominator in surds
5. Observing the connection between terms for each side (eg if the left side has a plus sign connecting two terms and the right side does not contain one, but instead contains a product of terms, the priority is to eliminate the plus signs)
6. Usage of basic algebraic-like identities such as a^2 - b^2 = (a + b) (a - b)
There are other techniques too, which you will face when proving such identities.
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