Prove that sin²α + cos²α = 1 and (1)/(cos^(2)\alpha ) = 1 + tan²α

Question

Prove that sin²α + cos²α = 1 and
`(1)/(cos^(2)\alpha )` = 1 + tan²α

Answer 1

See Explanation

Explanation:

`{ \sin}^(2) \alpha + { \cos}^(2) \alpha = 1 \\ \\ LHS = { \sin}^(2) \alpha + { \cos}^(2) \alpha \\ \\ = \cancel {\sin}^(2) \alpha + { 1 -\cancel{ \sin}}^(2) \alpha \\ \\ = 1 \\ \\ = RHS \\ \\ \\ \frac{1}{ { \cos}^(2) \alpha } = 1 + { \tan}^(2) \alpha \\ \\ LHS = \frac{1}{ { \cos}^(2) \alpha } \\ \\ = { \sec}^(2) \alpha \\ \\ = 1 + { \tan}^(2) \alpha \\ \\ = RHS`