use trigonometric and hyperbolic identities to solve and simplify the following:

4sin^2 theta +5cos theta =5
3tan^2 theta+5=7sec theta
prove that cosh^2x-sinh^2x=1
prove that sinh2x=2sinhxcoshx

any help with this would be much appreciated

thanks

Andy

1. Use the pythagorean identity,

You may use a substitution like cos A = x, then solve like a quadratic, plug back the trig function and solve for the angle.

2. Use another identity here. and proceed with the same method as above.

Sorry, I have not yet done hyperbolic identities. :(

Hi
I don't understand trig identities at all please could you explain it step by step

Thanks

Andy

Well, it's just like a substitution.

Ok, I'll show you the first one:





I'll use those two.





Are you okay with that?
If yes, then watch this:



Did you see? I replaced the sin^2 theta by the identity that I derived from the Pythagorean identity.

Now, expand;



Simplify;



As I told you before, make use of a substitution, so as not to confuse yourself. Say, cos theta = x.



I'm keeping the terms I just substituted in brackets for you to see them.
Then solve like a quadratic. I guess you can do that. You'll have then:

4x - 1 = 0
x = 0.25

and

x - 1 = 0
x = 1

From those two, replace them by the initial trigonometric ratio:

and

It's simple then. Grab your calculator, find the corresponding two values for each equation.

You should have: x = 0, 60, 300 or 360 (for x between 0 and 360 degrees inclusive)

These can be proved by simply substituting for the definitions of cosh(x) and sinh(x):



So, for example:



you can take it from here. The last proof can demonstrated using the same technique.

This was a double post, ebaines. Just to let you know.

https://www.askmehelpdesk.com/math-s...ng-432701.html