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.