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

1. $( \operatorname { sin } ( \theta ) + \operatorname { cos } ( \theta ) ) ^ { 2 } = 1 + 2 \operatorname { sin } ( \theta ) \operatorname { cos } ( \theta )$ 

   Hint (a):

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

2. $\operatorname { tan } ( \theta ) + \operatorname { cot } ( \theta ) = \operatorname { sec } ( \theta ) \operatorname { csc } ( \theta )$ 

   Hint (b):

   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(b):

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

   Step 2(b):

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

   Find a common denominator and then add the two terms.

   Step 3(b):

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

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

   Step 4(b):

   Substitute in the identities for the two terms.
   What does ^$1/\cosθ$ equal? ^$1/\sinθ$?

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

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

   Hint (c):

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