Question

Are these identities true for 0º < A < 90º ? If not, for which values of A they are true?
a. sec² A − tan²A = 1 b. cosec²A − cot²A = 1

Answer 1

HELLO DEAR,

both the identities are true for every value of A

let us see,

a). sec²A - tan²A

put A = 0,

⇒sec²0 - tan²0

⇒1 - 0 = 1

put A = 30°

⇒sec²30 - tan²30

⇒(2/√3)² - (1/√3)²

⇒4/3 - 1/3

⇒(4 - 1)/3

⇒3/3 = 1

similarly, sec²A - tan²A is Also valid for A = 0,30,60,90,...

b).cosec²A - cot ²A

put A = 0

cosec²0 - cot²0

⇒1  - 0 = 1

Put A = 30

⇒cosec²30 - cot²30

⇒(2)² - (√3)²

⇒4 - 3 = 1

Similarly, cosec²A - cot²A is also valid for A = 0,30,60,90,....

I HOPE ITS HELP YOU DEAR,

THANKS

Answer 2

Hi ,

i ) sec²A - tan²A = 1

A = 0° => sec² 0° - tan² 0° = 1² - 0² = 1

A = 30° => sec²30° - tan²30°

= ( 2/√3 )² - ( 1/√3 )²

= 4/3 - 1/3

= 3/3 = 1

A = 45° => sec² 45° - tan² 45°

= ( √2 )² - 1²

= 2 - 1

= 1

A = 60° => sec²60° - tan² 60°

= 2² - ( √3 )²

= 4 - 3

= 1

A = 90° => sec² 90° - tan² 90°

is not defined

Therefore ,

It is true for 0° ≤ A < 90°

ii ) cosec²A - cot² A = 1

A = 0° => cosec² 0° - cot² 0°

= undefined

A = 30° => cosec² 30° - cot² 30°

= 2² - ( √3 )²

= 4 - 3

= 1

A = 45° => cosec² 45° - cot² 45°

= ( √2 )² - 1²

= 2 - 1

= 1

A = 60° => cosec² 60° - cot² 60°

= ( 2/√3 )² - ( 1/√3 )²

= 4/3 - 1/3

= 3/3

= 1

A = 90° => cosec² 90° - cot² 90°

= 1² - 0²

= 1

Therefore ,

It is true for 0° < A ≥ 90°

I hope this helps you.

: )