Check all of the available options
⇒ First option sin²α = 1 + cos²α
Use first pythagorean identity to solve whether the equation is true
(Solve for sin²α)
sin²α + cos²α = 1
sin²α = 1 - cos²α
The result is different from the first option, so the first option is not valid
⇒ Second option 1 = sec²α - tan²α
Use second pythagorean identity (solve for 1)
tan²α + 1 = sec²α
1 = sec²α - tan²α
The result is the same, thus the second option is valid
⇒ Third option cot²α = csc²α - 1
Use third pythagorean identity (solve for cot²α)
1 + cot²α = csc²α
cot²α = csc²α - 1
The result is the same, thus the third option is valid
⇒ Fourth option 1 - tan²α = -sec²α
Use second pythagorean identity
tan²α + 1 = sec²α (multiply by -1)
-tan²α - 1 = -sec²α
-1 - tan²α = -sec²α
The result is different from the option, so the fourth option is not valid
⇒ Fifth option -cos²α = sin²α - 1
Use first pythagorean identity
sin²α + cos²α = 1
cos²α = 1 - sin²α (multiply by -1)
-cos²α = -1 + sin²α
-cos²α = sin²α - 1
The result is the same, thus the fifth option is valid
ANSWER: second, third, fifth option
