Question

Simplify (cos^2a - cot^2a)/(sin^2a - tan^2a)

Answer 1

To simplify the expression (cos^2a - cot^2a)/(sin^2a - tan^2a), we can apply trigonometric identities to rewrite and simplify the expression.

Step-by-step explanation:

To simplify the expression (cos^2a - cot^2a)/(sin^2a - tan^2a), we can use the trigonometric identities. Recall that cot a = 1/tan a and cos^2 a = 1 - sin^2 a.

Using these identities, we can rewrite the expression as ((1 - sin^2 a) - (1/tan^2 a))/(sin^2 a - (1 - sin^2 a)).

Further simplifying, we get (1 - sin^2 a - 1/tan^2 a)/(2 sin^2 a - 1).

Answer 2

The simplified expression is sec^2a

Step-by-step explanation:

We can start by using the trigonometric identities:

cot^2 a + 1 = csc^2 a

tan^2 a + 1 = sec^2 a

Using these identities, we can rewrite the expression as:

(cos^2 a - cot^2 a)/(sin^2 a - tan^2 a)

= (cos^2 a - (csc^2 a - 1))/(sin^2 a - (sec^2 a - 1))

= (cos^2 a - csc^2 a + 1)/(sin^2 a - sec^2 a + 1)

Now we can use the identity:

sin^2 a + cos^2 a = 1

to rewrite the expression further:

= (1/sin^2 a - 1/sin^2 a cos^2 a)/(1/cos^2 a - 1/cos^2 a sin^2 a)

= (1 - cos^2 a)/(sin^2 a - sin^2 a cos^2 a)

= sin^2 a / sin^2 a (1 - cos^2 a)

= 1 / (1 - cos^2 a)

= sec^2 a

Therefore, the simplified expression is sec^2 a.