How do you find the exact value of cos 36° using the sum and difference, double angle, or half-angle formulas?

Double angle formulas are used to express the trigonometric ratios of double angles (2θ) in terms of trigonometric ratios of single angle (θ) and similarly, half-angle formulas are used to express the trigonometric ratios of half angles ((1/2)θ) in terms of trigonometric ratios of single angle (θ).

Answer: The exact value of cos 36° using the sum and difference, double angle, or half-angle formulas is cos 36° = (√5 +1) / 4

Let's look into the steps below.

Explanation:

Let X = 18°

Therefore, 5X = 90°

⇒ 2X + 3X = 90°

⇒ 2X = 90° – 3X

Taking the sin on both sides, we get

sin 2X = sin (90˚ – 3X)

sin 2X = cos 3X  [Since, sin (90 - y) = cos y] -------------- (1)

We know that, according to the formula of multiple angles,

sin2x = 2 cos x sin x ------------ (2)

cos3x = 4cos³x – 3cosx -------------- (3)

Substituting (2) and (3) in (1) we get,

⇒ 2 sin X cos X = 4 cos³X – 3 cos X

⇒ 2 sin X cos X – 4 cos³X + 3 cos X = 0

⇒ cos X (2 sin X – 4 cos²X + 3) = 0

Divide both sides by cos X we get,

⇒ 2 sin X – 4 cos²X + 3 = 0

⇒ 2sinX – 4 (1 – sin²X) + 3 = 0 [Since, sin²y + cos²y = 1]

⇒ 4sin²X + 2sinX – 1 = 0

Let's use the quadratic formula to calculate the roots of the above quadratic equation

For a quadratic equation ax² +bx +c = 0, the values of root x will be

x = [−b ± √(b²−4ac)] / 2a

In the equation, 4sin²X + 2sinX – 1 = 0 we have

a = 4, b = 2, c = -1

Therefore,

sin X = [−2 ± √2² −4(4)(−1)] / 2(4)

⇒ sinX = [−2 ± √4+16] / 8

⇒ sinX = [−2 ± 2√5] / 8

⇒ sinX = (−1 ± √5) / 4

We will consider sinX = (−1 + √5) /4 because sin 18° lies in the first quadrant and it is always grater than 0.

Therefore,

sin 18° = sin X = −1 + √5/4

Now, cos 36° = cos (2×18°)

⇒ cos 36° = 1 – 2 sin²18° [ Since, cos 2x = 1 - 2 sin²x]

⇒ cos 36° = 1 – 2[(−1 + √5) / 4]²

⇒ cos 36° = [16 − 2 (5+1−2√5)] /16

⇒ cos 36° = (4 + 4√5) /16

⇒ cos 36° = (√5+1) /4

Thus, cos 36 = (√5+1) /4