We know cosec x= sec (90-x) so cosec 72=sec 18 now cosec ^2 72-tan^2 18= sec ^2 18-tan^2 18 we know sec x-tam x=1 so sec ^2 18- tan^2 18=1

Evaluation of (cosec^2 72 - tan^2 18)

Definition of Trigonometric Functions
Trigonometric functions are ratios of the sides of a right triangle. The six basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

Reciprocal Trigonometric Functions
The reciprocal trigonometric functions are formed by taking the reciprocal of the basic trigonometric functions. The reciprocal functions are cosecant, secant, and cotangent.

Derivation of Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables defined on the domain of the functions. Trigonometric identities are derived using the trigonometric functions and algebraic manipulations. One of the most basic identities is the Pythagorean identity, which states that sin^2 x + cos^2 x = 1.

Evaluation of (cosec^2 72 - tan^2 18)
To evaluate (cosec^2 72 - tan^2 18), we must first simplify each term using trigonometric identities. Recall that cosecant is the reciprocal of sine and tangent is the ratio of sine and cosine.

cosec^2 72 = 1/sin^2 72
Using the identity 1 + cot^2 x = cosec^2 x, we can write:
1 + cot^2 72 = cosec^2 72
cot^2 72 = cosec^2 72 - 1
Substituting this into our original expression, we get:
1/sin^2 72 - tan^2 18
= 1/sin^2 72 - (sin^2 18/cos^2 18)
Using the identity sin^2 x + cos^2 x = 1, we can write:
1/sin^2 72 - (sin^2 18/cos^2 18)
= 1/sin^2 72 - (1 - cos^2 18)/cos^2 18
= 1/sin^2 72 - 1/cos^2 18 + tan^2 18

Now, using the identity 1 + tan^2 x = sec^2 x, we can write:
1/sin^2 72 - 1/cos^2 18 + tan^2 18
= 1/sin^2 72 - 1/cos^2 18 + (sec^2 18 - 1)
= 1/sin^2 72 - 1/cos^2 18 + sec^2 18 - 1
= (1 - cos^2 72)/sin^2 72 + (1 - sin^2 18)/cos^2 18
= (sin^2 72)/sin^2 72 + (cos^2 18)/cos^2 18 - cos^2 72/sin^2 72
= 1 + 1 - cos^2 72/sin^2 72
= 2 - cot^2 72

Therefore, (cosec^2 72 - tan^2 18) = 2 - cot