Trig Question

Question

Prove sec²x-1=sec²xsin²x, justify significant steps. (state identities used)

Answer 1

sec²x - 1 = (1/cos²x) -1

 

              = (1/cos²x) - (cos²x/cos²x) = (1 - cos²x)/cos²x

 

                                                       = sin²x/cos²x

 

                                                       = (1/cos²x)(sin²x)

 

                                                       = sec²x sin²x

 

Answer 2

There is a Pythagorean relationship that states that:

 

1 + tan² Θ = sec² Θ

 

Therefore, tan² Θ = sec² Θ - 1

 

sec² x - 1 = tan² x

 

The secant and cosine functions are reciprocals of each other

 

sec² Θ = 1/(cos² Θ)

 

Hence, sec² x = 1/(cos² x)

 

Thus, we can re-write the right-hand side of the equation as:

 

[1/(cos² x)] * sin² x

 

= (sin² x)/(cos² x)

 

= [(sin x)/(cos x)]²

 

= (tan x)²

 

= tan² x

 

This demonstrates that the left-hand side of the equation is equal to the right-hand side.

Answer 3

sec²x - 1 = sec²x·sin²x

 

Now sec²x = 1/cos²x,  Substituting:

 

     (1/cos²x) - 1 = sin²x/cos²x

 

Multiply both sides by cos²x:

 

    1 - cos²x = sin²x

    1 = cos²x + sin²x 

 

Which is the Pythagorean Identity, known to be true.