Use identities to find the exact value of the trigonometric expression below.

1/cos2 * π/16 - 1/cot2 * π/16

PS: in words, the problem is, one divided by cosine-squared, then beside the cosine-square (still in the denominator) is pi/16. Minus one divided by cotangent-squared, then beside the cotangent-squared (still in the denominator) is pi dived by 16.) please show steps so that I can learn this.

let x = π/6 for easier typying

so your 1/cos2 * π/16 - 1/cot2 * π/16

= 1/cos^2 x - 1/cot^2 x
= sec^2 x - tan^2 x
since sec^2 x = 1 + tan^2 x ,
= 1 + tan^2 x - tan^2 x
= 1

To simplify the trigonometric expression, we will use some trigonometric identities.

First, let's simplify the expression 1/cos²(π/16):

1/cos²(π/16) can be rewritten as sec²(π/16) using the reciprocal identity: sec(θ) = 1/cos(θ).

Next, let's simplify the expression 1/cot²(π/16):

1/cot²(π/16) can be rewritten as tan²(π/16) using the reciprocal identity: tan(θ) = 1/cot(θ).

Now, we have:

sec²(π/16) - tan²(π/16)

Using the Pythagorean identity: sec²(θ) = 1 + tan²(θ), we can rewrite the expression as:

1 + tan²(π/16) - tan²(π/16)

The tan²(π/16) terms cancel out, giving us:

1 + 0

Therefore, the exact value of the trigonometric expression is 1.