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

prove that cosh^2x-sinh^2x=1


They can be expressed in terms of e. This stuff should be in your calc book. Just use them.

cosh(x)=\frac{e^{x}+e^{-x}}{2}

sinh(x)=\frac{e^{x}-e^{-x}}{2}

Now, apply these identites and do the algebra. You should get 1.

4sin^2 theta +5cos theta =5

Let's use an identity and try a substitution, then turn it into a quadratic.

4sin^{2}(t)+5cos(t)=5

4(1-cos^{2}(t))+5cos(t)=5

4-4cos^{2}(t)+5cos(t)=5

-4cos^{2}(t)+5cos(t)-1=0

Now, let u=cos(t)

-4u^{2}+5u-1=0

Solve the quadratic and resub.

Please, do not double post your questions.

https://www.askmehelpdesk.com/math-sciences/trigonometric-hyperbolic-identities-432694.html