(sec^(2)4x-1)/(sec^(2)4x)=sin^(2)4x

Question

please verify with detailed instructions. Thank you and have a great day.

Answer 1

Recall the identity 1 + tan²θ = sec²θ.  So, sec²θ - 1 = tan²θ.

 

Therefore,  (sec²(4x) - 1)/ sec²(4x) = tan²(4x)/sec²(4x)

 

                                                     = [sin²(4x)/cos²(4x)] ÷ [1/cos²(4x)]

 

                                                     = [sin²(4x)/cos²(4x)] • cos²(4x)

 

                                                     = sin²(4x) 

 

Answer 2

Let u = 4x to make it look simpler.

 

(sec²u - 1) / sec²u = (1/cos²u - 1) / (1/cos²u)

 

Multiply and divide by cos²u and get

 

(sec²u - 1) / sec²u  = (1 - cos²u) / 1 = 1 - cos²u

 

Use the identity

 

cos²u + sin²u = 1 ⇒ 1 - cos²u = sin²u

 

Then

 

(sec²u - 1) / sec²u = sin²u

 

[sec²(4x) - 1] / sec²(4x) = sin²(4x)