1. You have 11 Identities to prove
You have 11 Identities to prove. Please write them in your notebook or on a sheet of paper. As you proceed through this PPT, you may click for a hint or to see the solution. Even if you peek at the solution, you must still add the corresponding reasons to go with each statement.
IMPORTANT: I offer a sample solution. You may discover alternative solutions some shorter and some longer.

2. 1) Prove: 𝑐𝑜𝑡𝐴𝑠𝑒𝑐𝐴𝑠𝑖𝑛𝐴=1 CLICK FOR HINT
Work with left side; convert to sin and cos.
CLICK FOR HINT
CLICK FOR SOLUTION
= 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 ∙ 1 𝑐𝑜𝑠𝐴 ∙ 𝑠𝑖𝑛𝐴 1 =1

3. 2) Prove: 𝑐𝑜𝑠 2 𝐴 𝑠𝑒𝑐 2 𝐴−1 = 𝑠𝑖𝑛 2 𝐴 CLICK FOR HINT
Work with left side; use a Pythag ID.
CLICK FOR SOLUTION
= 𝑐𝑜𝑠 2 𝐴( 𝑡𝑎𝑛 2 𝐴)
= 𝑐𝑜𝑠 2 𝐴 1 ∙ 𝑠𝑖𝑛 2 𝐴 𝑐𝑜𝑠 2 𝐴
= 𝑠𝑖𝑛 2 𝐴

4. 3) Prove: 𝑠𝑖𝑛𝜃 𝑐𝑠𝑐𝜃−𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠 2 𝜃 CLICK FOR HINT
Work with left side; convert to sin and cos.
CLICK FOR HINT
=𝑠𝑖𝑛𝜃 1 𝑠𝑖𝑛𝜃 −𝑠𝑖𝑛𝜃
CLICK FOR SOLUTION
=1− 𝑠𝑖𝑛 2 𝜃
= 𝑐𝑜𝑠 2 𝜃

5. 4) Prove: 𝑐𝑠𝑐𝐴−𝑐𝑜𝑠𝐴𝑐𝑜𝑡𝐴=𝑠𝑖𝑛𝐴 CLICK FOR HINT
Work with left side; convert to sin and cos.
CLICK FOR HINT
CLICK FOR SOLUTION
= 1 𝑠𝑖𝑛𝐴 −𝑐𝑜𝑠𝐴∙ 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴
= 1− 𝑐𝑜𝑠 2 𝐴 𝑠𝑖𝑛𝐴
= 𝑠𝑖𝑛 2 𝐴 𝑠𝑖𝑛𝐴
=𝑠𝑖𝑛𝐴

6. 5) Prove: 𝑠𝑒𝑐𝛽−𝑡𝑎𝑛𝛽 𝑠𝑒𝑐𝛽+𝑡𝑎𝑛𝛽 =1 CLICK FOR HINT
Work with left side; FOIL and Pythag ID.
CLICK FOR HINT
CLICK FOR SOLUTION
= 𝑠𝑒𝑐 2 𝛽− 𝑡𝑎𝑛 2 𝛽 =1

7. 6) Prove: 1−𝑐𝑜𝑠𝑥 1+𝑠𝑒𝑐𝑥 𝑐𝑜𝑠𝑥= 𝑠𝑖𝑛 2 𝑥 CLICK FOR HINT
Work with left side; FOIL and simplify.
CLICK FOR SOLUTION
= 1+𝑠𝑒𝑐𝑥−𝑐𝑜𝑠𝑥−𝑐𝑜𝑠𝑥𝑠𝑒𝑐𝑥 𝑐𝑜𝑠𝑥
= 1+ 1 𝑐𝑜𝑠𝑥 −𝑐𝑜𝑠𝑥−𝑐𝑜𝑠𝑥 1 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑥
=𝑐𝑜𝑠𝑥+1− 𝑐𝑜𝑠 2 𝑥−𝑐𝑜𝑠𝑥
=1− 𝑐𝑜𝑠 2 𝑥
= 𝑠𝑖𝑛 2 𝑥

8. 7) Prove: 𝑠𝑒𝑐 2 𝜃−1 𝑐𝑠𝑐 2 𝜃−1 =1 CLICK FOR HINT CLICK FOR SOLUTION
Work with left side; use Pythag IDs.
CLICK FOR SOLUTION
= 𝑡𝑎𝑛 2 𝜃 𝑐𝑜𝑡 2 𝜃 =1

9. 8) Prove: 𝑐𝑜𝑡 2 𝜃 1+𝑐𝑠𝑐𝜃 +𝑠𝑖𝑛𝜃𝑐𝑠𝑐𝜃=𝑐𝑠𝑐𝜃 CLICK FOR HINT
Work with left side; convert to sin and cos.
CLICK FOR SOLUTION
= 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛 2 𝜃 𝑠𝑖𝑛𝜃 +𝑠𝑖𝑛𝜃 1 𝑠𝑖𝑛𝜃
= 𝑐𝑜𝑠 2 𝜃+ 𝑠𝑖𝑛 2 𝜃+𝑠𝑖𝑛𝜃 𝑠𝑖𝑛 2 𝜃+𝑠𝑖𝑛𝜃
= (1+𝑠𝑖𝑛𝜃) 𝑠𝑖𝑛𝜃(1+𝑠𝑖𝑛𝜃)
= 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛 2 𝜃 𝑠𝑖𝑛𝜃+1 𝑠𝑖𝑛𝜃 +1
= 1 𝑠𝑖𝑛𝜃
= 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛 2 𝜃 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃+1 +1
=𝑐𝑠𝑐𝜃
= 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛 2 𝜃+𝑠𝑖𝑛𝜃 +1

10. 9) Prove: 𝑡𝑎𝑛 2 𝑥 𝑠𝑒𝑐𝑥+1 +1=𝑠𝑒𝑐𝑥 CLICK FOR HINT
Work with left side; common denom.
= 𝑡𝑎𝑛 2 𝑥 𝑠𝑒𝑐𝑥+1 + 𝑠𝑒𝑐𝑥+1 𝑠𝑒𝑐𝑥+1
CLICK FOR SOLUTION
= 𝑡𝑎𝑛 2 𝑥+𝑠𝑒𝑐𝑥+1 𝑠𝑒𝑐𝑥+1
= 𝑠𝑒𝑐 2 𝑥−1+𝑠𝑒𝑐𝑥+1 𝑠𝑒𝑐𝑥+1
= 𝑠𝑒𝑐 2 𝑥+𝑠𝑒𝑐𝑥 𝑠𝑒𝑐𝑥+1
= 𝑠𝑒𝑐𝑥(𝑠𝑒𝑐𝑥+1) 𝑠𝑒𝑐𝑥+1
=𝑠𝑒𝑐𝑥

11. 10) Prove: 𝑐𝑜𝑠 3 𝑦+𝑐𝑜𝑠𝑦 𝑠𝑖𝑛 2 𝑦=𝑐𝑜𝑠𝑦 CLICK FOR HINT
Work with left side; factor.
CLICK FOR SOLUTION
=𝑐𝑜𝑠𝑦( 𝑐𝑜𝑠 2 𝑦+ 𝑠𝑖𝑛 2 𝑦)
=𝑐𝑜𝑠𝑦(1)
=𝑐𝑜𝑠𝑦

12. 11) Prove: 𝑠𝑒𝑐𝑦+𝑐𝑠𝑐𝑦 1+𝑡𝑎𝑛𝑦 =𝑐𝑠𝑐𝑦 CLICK FOR HINT
Work with left side; convert to sin and cos.
CLICK FOR SOLUTION
= 1 𝑐𝑜𝑠𝑦 + 1 𝑠𝑖𝑛𝑦 1+ 𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑦
= 𝑠𝑖𝑛𝑦+𝑐𝑜𝑠𝑦 𝑠𝑖𝑛𝑦𝑐𝑜𝑠𝑦 𝑐𝑜𝑠𝑦+𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑦
= 𝑠𝑖𝑛𝑦+𝑐𝑜𝑠𝑦 𝑠𝑖𝑛𝑦𝑐𝑜𝑠𝑦 𝑐𝑜𝑠𝑦 𝑠𝑖𝑛𝑦+𝑐𝑜𝑠𝑦
= 1 𝑠𝑖𝑛𝑦
=𝑐𝑠𝑐𝑦

13. Find your homework from last time
Find your homework from last time Finish whatever set (odds or evens) we didn’t already complete.