this question confused me can you please show me how its done.
Reduce (csc^2 x - sec^2 X) to an expression containing only tan x.
Try using the following identities:
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)
tan(x) = 1/cot(x)
csc x = 1/sin x
sec x = 1/cos x
tan x = 1/cot x
sin^2 x + cos^2 x = 1
1 + cot^2 x = csc^2 x
tan^2 x + 1 = sec^2 x
csc^2 x - sec^2 x
= 1 + cot^2 x - (1 + tan^2 x)
= cot^2 x - tan^2 x
= (1/tan^2 x) - tan^2 x
Is it correct??
Correct!
To reduce the expression (csc^2 x - sec^2 x) to an expression containing only tan x, we can use trigonometric identities.
First, let's recall the definitions of the trigonometric functions:
csc x = 1/sin x
sec x = 1/cos x
tan x = sin x / cos x
Using these definitions, we can express csc^2 x and sec^2 x in terms of sin x and cos x:
csc^2 x = (1/sin x)^2 = 1/sin^2 x
sec^2 x = (1/cos x)^2 = 1/cos^2 x
Now, substitute these expressions into the original expression:
(csc^2 x - sec^2 x) = (1/sin^2 x) - (1/cos^2 x)
Next, let's combine the two fractions by finding a common denominator. The common denominator is sin^2 x * cos^2 x, so we multiply the first fraction by cos^2 x and the second fraction by sin^2 x:
(csc^2 x - sec^2 x) = (cos^2 x/sin^2 x * cos^2 x) - (sin^2 x/cos^2 x * sin^2 x)
Simplifying further:
(csc^2 x - sec^2 x) = cos^2 x * cos^2 x/sin^2 x - sin^2 x * sin^2 x/cos^2 x
Now, recall the Pythagorean identity: sin^2 x + cos^2 x = 1. Rearranging this identity, we get:
cos^2 x = 1 - sin^2 x
Substituting this into the expression:
(csc^2 x - sec^2 x) = (1 - sin^2 x) * (1 - sin^2 x) / sin^2 x - sin^2 x * sin^2 x / (1 - sin^2 x)
Simplifying further:
(csc^2 x - sec^2 x) = (1 - sin^2 x)^2 / sin^2 x - sin^4 x / (1 - sin^2 x)
Expanding (1 - sin^2 x)^2 = 1 - 2sin^2 x + sin^4 x:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x + sin^4 x) / sin^2 x - sin^4 x / (1 - sin^2 x)
Combining the fractions under a common denominator:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x + sin^4 x - sin^4 x) / sin^2 x * (1 - sin^2 x)
Simplifying further:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) / sin^2 x * (1 - sin^2 x)
Now, recall the identity: 1 - sin^2 x = cos^2 x:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) / sin^2 x * cos^2 x
Finally, substitute the definition of tan x: sin x / cos x:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) / sin^2 x * (sin x / cos x)^2
Simplifying further:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) * (sin x)^2 / (sin^2 x * (cos x)^2)
Canceling out the common factors:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) * sin x / (cos x)^2
And that's the final expression containing only tan x:
(csc^2 x - sec^2 x) = (1 - 2sin^2 x) * sin x / (cos x)^2
I hope this explanation helps! Let me know if you have any further questions.