Prove that each of the following equations is an identity. In other words, prove that they are each true for all values for which the functions are defined.

1. $\left(\sin\theta+\cos\theta\right)^2=1+2\sin\theta \cos\theta$ 

   Hint (a):

   Start by expanding the left side of the equation.
   You might have to rearrange the equation to see ways to simplify it after you expand it.

   Step 1(a):

   Change the trigonometric identities on the left side of the equation into the ratios they represent.

   $\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=$

   Step 2(a):

   Find a common denominator and then add the two terms.

   $\frac{\sin^{2}\theta}{\cos\theta\sin\theta}+\frac{\cos^{2}\theta}{\cos\theta\sin\theta}=$

   Step 3(a):

   Remember that $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$, and rewrite the equation.

   $\frac{1}{\cos\theta\sin\theta}=$

   Step 4(a):

   Substitute in the identities for the two terms.
   What does $\frac{1}{\cos\theta}$ equal? $\frac{1}{\sin\theta}$?

2. $\tan\theta+\cot\theta=\sec\theta\csc\theta$ 

   Hint (b):

   Substitute the trig identities in first, then expand by multiplying the two terms together.
   Remember to use Giant Ones to simplify when you can.

3. $\left(\tan\theta\cos\theta\right)\left(\sin^2\theta+\frac{1}{\sec^2\theta}\right)=\sin\theta$ 

   Hint (c):

   Use methods from both parts (a) and (b).