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See below for the trigonometrical identities. All these identities are given in the O level A Maths exam. However, one should be able to remember them after sometime if one has put in enough effort to practise questions on trigonometry. I recall last time as a student, I was very interested in mathematics and I practised my trigonometry so well that I actually remembered all these identities. Thus, I did not refer to them during my actual O level A maths exam as these identities were already in my head. There was no need for me to keep flipping to the formula page and this resulted in so much time saved and lesser stress during the exam. So, the more passionate one is into something, the harder it is for him/ her to forget that thing. It becomes a part of him/ her. Of course, we have to be careful what thing we should be passionate about as not all things are desirable in life.
Proving trigonometrical identities (E.g. 1)
Tip: We start from the LHS in this example since there are more things there to work with to simplify to the expression on the RHS.
We are adding the two fractions together. This thinking is rather instinctive since there are two fractions and we just have to add them to continue the proving.
We are using the identity sin²Ө + cos²Ө = 1 here in the process of proving.
We are adding the two fractions together. This thinking is rather instinctive since there are two fractions and we just have to add them to continue the proving.
We are using the identity sin²Ө + cos²Ө = 1 here in the process of proving.
Proving trigonometrical identities (E.g. 2)
We start from the LHS in this example as there are more things there to work with to prove into RHS.
The trick we use here is algebraic factorisation rule a² - b² = (a+b)(a-b). As we can see, it is important to know your algebraic rules very well as algebra is one of the cornerstones of maths. It is related to many branches of maths. In doing proving questions, we should always be flexible in our thinking as even algebraic rules can also be used in the process of proving.
The trick we use here is algebraic factorisation rule a² - b² = (a+b)(a-b). As we can see, it is important to know your algebraic rules very well as algebra is one of the cornerstones of maths. It is related to many branches of maths. In doing proving questions, we should always be flexible in our thinking as even algebraic rules can also be used in the process of proving.
Proving trigonometrical identities (E.g. 3)
In this example, we start from LHS and use the two identities sec²Ө = 1 + tan²Ө and cosec²Ө = 1 + cot²Ө. Actually, we can also start from RHS to prove into the LHS. Both of the two sides each has enough "stuff" there to manipulate to the other side, so in this example it does not matter which side we start from. Try it for yourself to prove from RHS to LHS and see that it should work also.
Proving trigonometrical identities (E.g. 4)
In this example, there is an important "trick" employed. This example calls this the "1" trick. We multiply the whole expression by a fraction that has the same numerator and denominator. This example uses the "1" trick twice in multiplying the fractions sinӨ / sinӨ and also (1 - cosӨ) / (1 - cosӨ) to the expression while in the process of proving. Learn this trick if it is not familiar to you as this is a very super wonderful trick which can make proving so much easier whenever this trick can be used.
Proving trigonometrical identities (E.g. 5)
The same "1" trick used again. The fraction which is multiplied to the expression is (1 + cosӨ) / (1 + cosӨ). Further onwards in the proving, we can also employ another "trick" which is to divide both the numerator and denominator of the fraction expression by sinӨ. We are not doing anything mathematically illegal here. We can simplify both the numerator and denominator of a fraction expression by multiplying by the same thing (the "1" trick) or dividing by the same thing (actually also the "1" trick again).
However, please NOTE that we cannot add or subtract the same thing on both the numerator and denominator. This same thing "trick" DOES NOT APPLY FOR ADDITION AND SUBTRACTION of same thing on numerator and denominator of a fraction expression! This "1" trick ONLY APPLIES FOR MULTIPLICATION AND DIVISION of same thing to numerator and denominator of a fraction expression!
However, please NOTE that we cannot add or subtract the same thing on both the numerator and denominator. This same thing "trick" DOES NOT APPLY FOR ADDITION AND SUBTRACTION of same thing on numerator and denominator of a fraction expression! This "1" trick ONLY APPLIES FOR MULTIPLICATION AND DIVISION of same thing to numerator and denominator of a fraction expression!
Proving trigonometrical identities (E.g. 6)
Proving trigonometrical identities (E.g. 7)
Proving trigonometrical identities (E.g. 8)
The beauty of the "1" trick employed again here. In this example, the complicated fraction expression is simplified by multiplying it by (sinӨcosӨ) / (sinӨcosӨ).
Proving trigonometrical identities (Double angles E.g. 1)
In this example, the double angle identity cos2A = 1 -2sin²A and sin2A = 2sinAcosA are being used. Note here that when doing proving, one must have a clear direction where you are headed. Since the RHS is tanA which is sinA / cosA and we need sinA at the numerator, so we purposely choose to use cos2A = 1 - 2sin²A to change the numerator on the LHS.
I hope to impress upon you that what identity to use is very intentional and one must be quick to observe what we are trying to prove to on the other side in order to employ the fastest way to prove to the other side. We must always OBSERVE, OBSERVE and OBSERVE again and again carefully so that we use the right identity and not just randomly use any identity to avoid wasting time and effort and still not able to prove to the other side.
One common mistake is that students just jump into the proving straight away and hope things will turn out fine as long any random identity is used in the process of proving. The A grade student OBSERVE VERY CAREFULLY what he/ she is trying to get to the other side in the proving and think CAREFULLY before ACTING accordingly by purposeful using of appropriate identity.
I hope to impress upon you that what identity to use is very intentional and one must be quick to observe what we are trying to prove to on the other side in order to employ the fastest way to prove to the other side. We must always OBSERVE, OBSERVE and OBSERVE again and again carefully so that we use the right identity and not just randomly use any identity to avoid wasting time and effort and still not able to prove to the other side.
One common mistake is that students just jump into the proving straight away and hope things will turn out fine as long any random identity is used in the process of proving. The A grade student OBSERVE VERY CAREFULLY what he/ she is trying to get to the other side in the proving and think CAREFULLY before ACTING accordingly by purposeful using of appropriate identity.
Proving trigonometrical identities (Double angles E.g. 2)
Proving trigonometrical identities (Double angles E.g. 3)
Proving trigonometrical identities (Half-angles E.g. 1)
This example involves half-angles in the proving. Actually, this is still using the double angle identity to help in the proving, so I will rather still see half-angles as similar to double angles concept.
Proving trigonometrical identities (Addition formulae E.g. 1)
Addition formulae used in beginning of proving. After that, the "1" trick used in this example by dividing both numerator and denominator of fraction expression by the same thing which is cosAcosB. Again, beauty of the "1" trick being used here.