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 = ±\(\sqrt{\frac{1  cos 2A}{1 + cos 2A}}\)
11 and 12 Grade Math
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