Question

Which of the following is an identity? A. sin2x sec2x + 1 = tan2x csc2x B. sin2x - cos2x = 1 C. (cscx + cotx)2 = 1 D. csc2x + cot2x = 1

Edit: no one answered but I figured out that the right answer is A

Answer 1

Answer: A is the correct answer.

Explanation:

Just did the question, followed OP's advice since no one answered. Good luck.

Answer 2

There are three 'Pythagorean' identities that we can look at and they are

sin²(x) + cos²(x) = 1
tan²(x) + 1 = sec²(x)
1 + cot²(x) = csc²(x)

We can start by checking each option to see which one would give us any of the 'Pythagorean' identities as its simplest form

Option A:

sin²(x) sec²(x) + 1 = tan²(x) csc²(x)

Rewriting sec²(x) as 1/cos²(x)
Rewriting tan²(x) as sin²(x)/cos²(x)
Rewriting csc²(x) as 1/sin²(x)

We have


`sin^(2)(x)[ (1)/( cos^(2)(x) )]+1=[ ( sin^(2)( x))/( cos^(2) (x))][ (1)/( sin^(2)(x) ) ]`

`[( sin^(2)(x) )/( cos^(2)(x) ) ]+1= (1)/( cos^(2)(x) )`

`tan^(2)(x)+1= sec^(2)(x)`

Option B:

sin²(x) - cos²(x) = 1

This expression is already in the simplest form, cannot be simplified further

Option C:

[ csc(x) + cot(x) ]² = 1

Rewriting csc(x) as 1/sin(x)
Rewriting cot(x) as cos(x)/sin(x)

We have


`[ (1)/(sin(x))+ (cos(x))/(sin(x))] ^(2) =1`

`(1)/(sin^2(x))+2( (1)/(sin(x)))( (cos(x))/(sin(x)))+ (cos^2(x))/(sin^2(x))=1`
`csc^2(x)+2csc^2(x)cos(x)+cot^2(x)=1`

Option D:

csc²(x) + cot²(x) = 1

Rewriting csc²(x) as 1/sin²(x) and cot²(x) as cos²(x)/sin²(x)


`(1)/(sin^2(x))+ (cos^2(x))/(sin^2(x))=1`

`(1+cos^2(x))/(sin^2(x)) =1`

`1+cos^2(x)=sin^2(x)`

`1=sin^2(x)-cos^2(x)`

from our working out we can see that option A simplified into one of 'Pythagorean' identities, hence the correct answer