# Trigonometric Functions of A in Terms of cos 2A

We will learn how to express trigonometric functions of A in terms of cos 2A or trigonometric ratios of an angle A in terms of cos 2A.

We know the formula of cos 2A and now we will apply the formula to proof the below trigonometric ratio of multiple angle.

(i) Prove that: cos$$^{2}$$ A = $$\frac{1 + cos 2A}{2}$$  i.e., cos A = ±$$\sqrt{\frac{1 + cos 2A}{2}}$$

We know that, cos 2A = 2 cos^2 A - 1

⇒ cos$$^{2}$$ A = $$\frac{1 + cos 2A}{2}$$

i.e., cos A = ±$$\sqrt{\frac{1 + cos 2A}{2}}$$

(ii) Prove that: sin$$^{2}$$ A = $$\frac{1 - cos 2A}{2}$$  i.e., sin A = ±$$\sqrt{\frac{1 + cos 2A}{2}}$$

We know that, cos 2A = 1 - 2 sin^2 A

⇒ sin$$^{2}$$ A = $$\frac{1 - cos 2A}{2}$$

i.e., sin A = ±$$\sqrt{\frac{1 + cos 2A}{2}}$$

(iii) Prove that: tan$$^{2}$$ A = $$\frac{1 - cos 2A}{1 + cos 2A}$$  i.e., tan A = ±$$\sqrt{\frac{1 - cos 2A}{1 + cos 2A}}$$

We know that, tan$$^{2}$$ A = $$\frac{sin^{2} A}{cos^{2} A}$$

⇒ $$\frac{1 - cos 2A}{1 + cos 2A}$$

i.e., tan A = ±$$\sqrt{\frac{1 - cos 2A}{1 + cos 2A}}$$

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