The general method I execute in trying to verify trigonometric identities is to
1.) rewrite everything in terms of sin and cos
2.) sum the terms together, i do this for both sides of the identity
3.) now at this point if the identity hasn't been verified, the problem arises where both sides contain a single term, but one side's term is more complicated than the other.
Sometimes a greatest common factor is staring me in the face, here it can be easy to cancel down into the simpler side thus the identity is verified.
However, The problem I am concerned with is when the more complicated side's numerator and denominator do not contain a Greatest common factor, as they are right now. Simple example of this( i say simple but i spent a solid ten minutes on it lol )
[math]\frac{1-\sin^4(x)}{\cos^4(x)}=\frac{1+\sin^2(x)}{\cos^2(x)}[/math]
As we can see there is one term on both sides of the identity, yet one term is more complicated. this is the one i must reduce to the other.
DoTs works here;
[math]\frac{(1-\sin^2(x))(1+\sin^2(x))}{\cos^2(x)(1-\sin^2(x))}=\frac{1+\sin^2(x)}{\cos^2(x)}[/math]
So in the pursuit of verifying a trig identity ( assuming the identity is actually true) with this specific strategy, eventually
I am looking for a general method of doing that. Or is it all done by "experience" and "intuition" and "doing lots of problems"?
Now my progress in this problem is as follows:
- since i ensured i wrote everything in the identity in terms of sin and cos, i can ignore any identity that contains a trig term other than sin or cos or their natural powers. this leaves very few identities which is good.
- Analysing the differences in the respective numerators and respective denominators.
sign changes suggest the pythagorean identity must be used
differences of powers of exactly two(or perhaps any multiple of two) suggest the pythagorean identity is required as well.
- difference of two squares rule has also been helpful
any advice? thx
1.) rewrite everything in terms of sin and cos
2.) sum the terms together, i do this for both sides of the identity
3.) now at this point if the identity hasn't been verified, the problem arises where both sides contain a single term, but one side's term is more complicated than the other.
Sometimes a greatest common factor is staring me in the face, here it can be easy to cancel down into the simpler side thus the identity is verified.
However, The problem I am concerned with is when the more complicated side's numerator and denominator do not contain a Greatest common factor, as they are right now. Simple example of this( i say simple but i spent a solid ten minutes on it lol )
[math]\frac{1-\sin^4(x)}{\cos^4(x)}=\frac{1+\sin^2(x)}{\cos^2(x)}[/math]
As we can see there is one term on both sides of the identity, yet one term is more complicated. this is the one i must reduce to the other.
DoTs works here;
[math]\frac{(1-\sin^2(x))(1+\sin^2(x))}{\cos^2(x)(1-\sin^2(x))}=\frac{1+\sin^2(x)}{\cos^2(x)}[/math]
So in the pursuit of verifying a trig identity ( assuming the identity is actually true) with this specific strategy, eventually
it becomes a game of manipulating the numerator and denominator with the available body of trig identities until a GCF can be extracted.
I am looking for a general method of doing that. Or is it all done by "experience" and "intuition" and "doing lots of problems"?
Now my progress in this problem is as follows:
- since i ensured i wrote everything in the identity in terms of sin and cos, i can ignore any identity that contains a trig term other than sin or cos or their natural powers. this leaves very few identities which is good.
- Analysing the differences in the respective numerators and respective denominators.
sign changes suggest the pythagorean identity must be used
differences of powers of exactly two(or perhaps any multiple of two) suggest the pythagorean identity is required as well.
- difference of two squares rule has also been helpful
any advice? thx