I'm stuck and i don't know how to do this.
(4tan^2(θ)cot^2(θ)/csc^2(θ))+(4/tan^2(θ)+1)
Hello, deezeejoey!
Do you remember any of the basic identities?
$\displaystyle \frac{4\tan^2\!\theta\cot^2\!\theta}{\csc^2\!\thet a}+ \frac{4}{\tan^2\!\theta+1}$
$\displaystyle \text{The first fraction is: }\;\frac{4(\overbrace{\tan\theta\cot\theta}^{\text {This is 1}})^2}{\csc^2\!\theta} \;=\;\frac{4}{\csc^2\!\theta} \;=\;4\sin^2\!\theta$
$\displaystyle \text{The second fraction is: }\;\frac{4}{\underbrace{\tan^2\!\theta + 1}_{\text{This is }\sec^2\theta}} \;=\;\frac{4}{\sec^2\!\theta} \;=\;4\cos^2\!\theta$
$\displaystyle \text{The problem becomes: }\;4\sin^2\!\theta + 4\cos^2\!\theta \;=\;4\underbrace{(\sin^2\!\theta + \cos^2\!\theta)}_{\text{This is 1}} \;=\;4\cdot1 \;=\;4$