Question

If a square secant squared theta minus b square tan square theta is equal to c square then prove that sin square theta is equal to c square minus a square by c square minus b square

Answer 1

AN IDENTITY IS AN EQUALITY that is true for any value of the variable. (An equation is an equality that is true only for certain values of the variable.)

In algebra, for example, we have this identity:

(x + 5)(x − 5) = x2 − 25.

The significance of an identity is that, in calculation, we may replace either member with the other.  We use an identity to give an expression a more convenient form.  In calculus and all its applications, the trigonometric identities are of central importance.

On this page we will present the main identities.  The student will have no better way of practicing algebra than by proving them.  Links to the proofs are below.

Reciprocal identities

sin θ =     1  
csc θ       csc θ =     1  
sin θ cos θ =     1  
sec θ       sec θ =     1  
cos θ tan θ =     1  
cot θ       cot θ =     1  
tan θ

Proof

Again, in calculation we may replace either member of the identity with the other.  And so if we see "sin θ", then we may, if we wish, replace

  it with "   1  
csc θ";  and, symmetrically, if we see  "  1  
csc θ",

then we may replace it with "sin θ".