High school trig identities questions These are a few trig identities questions that I just can't figure out. They are from the Cambridge 3U book. 
I hate to put more than one question up at a time but I just can't figure any of them out, anyhow:


*

*Eliminate theta from the following pair of equations:


\begin{align}x &= \sin(\theta) - 3\cos(\theta)\\
y &= \sin(\theta) + 2\cos(\theta)\end{align}


*If $\tan(\theta) + \sin(\theta) = x$ and $\tan(\theta) - \sin(\theta) = y$, prove that $$x^4 + y^4 = 2xy(8 + xy)$$

*If $\dfrac a{\sin A}= \dfrac b{\cos A}$, show that $$\sin(A)\cos(A) = \frac{ab}{a^2 + b^2}$$

*If $\dfrac{a + b}{\text{cosec}(x)} = \dfrac{a - b}{\cot(x)}$, show that $$\text{cosec}(x)\cot(x) = \frac{a^2 - b^2}{4ab}$$
 A: *

*Put $a = \sin\theta$ and $b = \cos\theta$. Two equations in two variables $a$ and $b$. Solve for $a$ and $b$ and then use $a^2 + b^2 = 1$.

*Directly substitute $x$ and $y$ and check LHS and RHS.

*Square both sides and use $\sin^2A + \cos^2A = 1$.

*$\cos x$ is given. Using $\sin^2x + \cos^2x = 1$, compute $\sin x$ and then substitute in last equation.
There might be other simpler ways to solve these. But for now these methods will keep you going.
A: $4)\dfrac{a+b}{\csc x}=\dfrac{a-b}{\cot x}=\pm\sqrt{\dfrac{(a+b)^2-(a-b)^2}{\csc^2x-\cot^2x}}=?$
Observe that $\csc x,\cot x$ will have the same sign.
$(3)$ Can you use the method applied in $(4)$
$(2)$  Solve for $\tan\theta,\sin\theta$
Use $$\dfrac1{\sin^2\theta}-\dfrac1{\tan^2\theta}=\cdots=1$$
$(1)$  Solve for $\sin\theta,\cos\theta$
Use Fundamental Theorem of Trigonometry
A: Multiply the two fractions together the answer will be the the square of any of them , sin^2A+cos^2A=1
 ab/(sinAcosA)=a^2/sin^2A=b^2/cos^2A=a^2+b^2/1
Then make cross multiplication to get the answer
