Maths C3 - Trigonometry... Help??

You'll also notice why $\cot\theta \not \equiv \dfrac{1}{\tan\theta}$ from solving this equation.

I find it absurd how so many place say that it is an IDENTITY, when this question is a perfect example of why it is not.

Essentially yeah, but your second line wouldn't be correct because if I were to multiply both sides by $\tan \theta$ then I would be left with $1=0\cdot \tan \theta = 0 \Rightarrow 1=0$ which doesn't make sense.
Again, you made the same mistake by assuming $\cot \theta \equiv \frac{1}{\tan \theta}$. If you ignore line 2, then that working would be fine.

Also it would see more straight forward if from your fractorised form you went:

$\frac{\cos \theta}{\sin \theta}(1+\cot \theta)=0 \Rightarrow \cos \theta (1+\cot \theta)=0$

From multiplying both sides by sine.

Question for my own interest (not meant to be patronising and I don't know the answer to it) but how easily do you feel that you give up trying to solve a question, in particular for when you post here?

This is to kind of understand A. How helpful this is and B. Whether you can change your approach to questions slightly. I

In all honesty I must have spent a good hour trying to solve that one question I'm self-teaching Maths so when I'm stuck, I really get stuck! Then I start to lose faith in myself. Doesn't help that sometimes I feel like there are huge gaps in my knowledge. Also sometimes the Edexcel Endorsed textbook doesn't explain a particular topic too well, so for an amateur like me it's really difficult to get my head around some things

Do you ever use ExamSolutions? His explanations are excellent and go beyond what a textbook can tell you. There's a tutorial for every A Level topic.

I wish there was something like ExamSolutions back when I self-taught Further Maths. I used a textbook like you and spent a lot of time asking questions on this forum. Self-teaching can be really tough sometimes so I know how you feel.

Keep asking questions on TSR and that will definitely help. As you've probably noticed there are many excellent members in this forum who can explain things better than a textbook.

It is an identity. Do you claim that $1 + \tan^2 \theta = \sec^2 \theta$ isn't an identity because it fails for $\theta = \frac{\pi}{2}$?

It is an identity. Do you claim that $1 + \tan^2 \theta = \sec^2 \theta$ isn't an identity because it fails for $\theta = \frac{\pi}{2}$?

When solving difficult equations like...

$(sec \theta - cos \theta)^2 = tan \theta - sin^2 \theta$ .... for the interval... $0 \leq \theta \leq \pi$

Out of sin, cos, and tan how do you know which one it'll likely simplify down to?
I feel like I'm just doing this out of trial and error without an aim

When solving difficult equations like...

$(sec \theta - cos \theta)^2 = tan \theta - sin^2 \theta$ .... for the interval... $0 \leq \theta \leq \pi$

Out of sin, cos, and tan how do you know which one it'll likely simplify down to?
I feel like I'm just doing this out of trial and error without an aim

Most likely cosine as it dominates most of the equation, but you'd usually have to play around with it and see first.

It's not always possible (or realistic) to see a solution right to its end. Here a sensible thing to do would be to expand the brackets. One can see in advance that expanding the brackets simplifies things a lot.

Yes, I see there are actually quite a few cosines in the equation. Thank you


Yeah straight away I noticed that the squared bracket could be expanded because of the form $(a-b)^2 = a^2 -2ab=b^2$

But 10 lines worth of workings I still find myself stuck on this question

Yes, I see there are actually quite a few cosines in the equation. Thank you


Yeah straight away I noticed that the squared bracket could be expanded because of the form $(a-b)^2 = a^2 -2ab=b^2$

But 10 lines worth of workings I still find myself stuck on this question

$(\sec \theta - \cos \theta)^2 = \tan \theta - \sin^2 \theta$

$\Rightarrow \sec^2 \theta - 2 + \cos^2 \theta = \tan \theta - \sin^2 \theta$

$\Rightarrow \sec^2 \theta -2 + (\cos^2 \theta + \sin^2 \theta) = \tan \theta$

Can you see where to go from there? The $\sec^2 \theta$ can be expressed in terms of $\tan \theta$ and the bracket is your base identity. Working can be done in 5 lines max.

OMG!! How did I not see the... $cos^2 \theta + sin^2 \theta$??? I even had it written in that order!!

Why are these trig equations so difficult for me to get used to?

Because you don't do them as much before C3, you'll be fine

Seriously on the verge of giving up

Does anyone have any advice for these type of Trig Identity questions that I keep getting stuck on? There's got to be some systematic way of working through questions because I am just baffled I feel so useless!!

Seriously on the verge of giving up

Does anyone have any advice for these type of Trig Identity questions that I keep getting stuck on? There's got to be some systematic way of working through questions because I am just baffled I feel so useless!!

There is no systematic way, this isn't GCSE maths anymore. You just have to be aware of all your identities and ultimately aim to get your equations in terms on a single trigonometric function, from which point they are easy to solve.

Just as RDK said, there is no systematic way. If there was, everyone would be coasting through every question.

I mean, it doesn't get anymore systematic than

1. Be as familiar as possible (and on the look out for) identities that you know. The main ones being $\sin^2+\cos^2=1$ and the two corresponding identities obtained by dividing this equation.

2. The whole goal with any problem with any new notation/function is getting rid of the new notation/function, which is usually achieved by isolating it. In the case of trig equations it's converting the equation into one in terms of just trig function.
Then it's easy to solve as normal. Keep that it mind and you should be fine.

And no it's not as black and white as you were thinking where you're supposed to decide what to convert to at the start - this isn't always possible. Just use the identities you know to simplify things and ultimately get the equation in terms of one trig function. I can guarantee the next time you get stuck, it will be because you either didn't apply one of the standard identities you know (purposefully of course) or you didn't make some sort of obvious simplification to the equation such as clearing fractions, expanding brackets etc. Or you just made a silly mistake lol.

I had a feeling that would be the case


Thank you. I've started getting into the habit of writing trig identities down as I go along so I can spot any simplifications whilst I'm staring at the equation. But why does it seem that these types of questions take me like 30 minutes each? I'm stuck on another question as we speak that's how much I'm struggling