# Prepare For Cosecant Exam

Introduction :

The definition of Cosecant (Csc)   is nothing but the function which is used to calculate the ratio of sides of the triangle. It is  also known as circular function. csc is a one kind of trigonometric functions. Csc of an angle is the ratio of hypotenuse and the length of the opposite side. In other words, the definition of Csc  is the reciprocal of sin.

In a right angle triangle,

Csc (x) =   hypotenuse       (or)        1
opposite side                 sin (x) .
Here x is an angle.

## Prepare for Properties of Csc (Cosecant) exam :

By the definition of csc,

csc x               = 1/sin x

csc (-x)            = - csc(x)

csc(ix)             = -i csc h x

Domain of csc :

Every numbers are   R – {(n∏ | n Z} ,  n is an integer.

Range of csc:

(-∞ , -1] U [1 , +∞)

Co function for csc :

csc x  = sec (90o - x).

I am planning to write more post on What is a Acute Angle with example, Equation of Ellipse. Keep checking my blog.

## Prepare for problems on cosecant exam

Prepare example 1:

Solve trigonometric expression:

[csc (a) cos2 a] / [1 +csc(a)]

Solution :

[csc (a) cos 2a] / [1 + csc (a)] = cos2a / [ sin a (1 + csc (a) ]   here ,   csc (a)= 1 / sin(a)

= cos 2a / [ sin a + 1 ]

= [ 1 - sin 2a ] / [ sin a + 1 ]     Here  cos2a by 1 - sin 2a ,

= [ (1 – sin a)(1 + sin  a) ] / [ sin a + 1 ]

[csc (a) cos2 a] / [1 +csc(a)]            = 1 – sin a

Prepare example 2:

Prove that, sec2x + csc2x = sec2x csc2x

Proof:       L.H.S. => sec2x + csc2x

=> (1 / cos2x) + (1 / sin2x)                      [Using Reciprocal Identity:secΘ = 1 / cosΘ ; cscΘ = 1 / sinΘ]

=> (sin2x + cos2x) / sin2x × cos2x          [Taking L.C.M. ]

=> 1 / sin2x × cos2x                               [Using Pythagorean Identity: sin2x + cos2x = 1]

=> 1 / sin2x × 1 / cos2x                          [Separating it Into Two Terms]

=> csc2x × sec2x                                   [Using Reciprocal Identity: 1 / sinΘ = cscΘ; 1 / cosΘ = secΘ]

=> sec2x csc2x

=> R.H.S.

sec2x + csc2x = sec2x × csc2x.