Question

please solve this for me ​

Answer 1

Answer:

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Step-by-step explanation:

We know that for two real numbers or angles A and B,

cos (A + B) = cos A cos B - sin A sin B

Now, putting B = A on both sides of the above formula we get,

cos (A + A) = cos A cos A - sin A sin A

⇒ cos 2A = cos² A - sin² A

⇒ cos 2A = cos² A - (1 - cos² A), [since we know that sin² θ = 1 - cos² θ]

⇒ cos 2A = cos² A - 1 + cos² A,

⇒ cos 2A = 2 cos² A - 1

⇒ cos 2A = 2 (1 - sin² A) - 1, [since we know that cos² θ = 1 - sin² θ]

⇒ cos 2A = 2 - 2 sin² A - 1

⇒ cos 2A = 1 - 2 sin² A

Note :-

(i) From cos 2A = 2 cos² A - 1 we get, 2 cos² A = 1 + cos 2A

and from cos 2A = 1 - 2 sin² A we get, 2 sin²A = 1 - cos 2A

(ii) In the above formula we should note that the angle on the R.H.S. is half of the angle on L.H.S. Therefore, cos 120° = cos² 60° - sin² 60°.

(iii) The above formulae is also known as double angle formulae for cos 2A.

Hope my solution helps you..

Answer 2

Step-by-step explanation:

We know that for two real numbers or angles A and B,

cos (A + B) = cos A cos B - sin A sin B

Now, putting B = A on both sides of the above formula we get,

cos (A + A) = cos A cos A - sin A sin A

⇒ cos 2A = cos² A - sin² A

⇒ cos 2A = cos² A - (1 - cos² A), [since we know that sin² θ = 1 - cos² θ]

⇒ cos 2A = cos² A - 1 + cos² A,

⇒ cos 2A = 2 cos² A - 1

⇒ cos 2A = 2 (1 - sin² A) - 1, [since we know that cos² θ = 1 - sin² θ]

⇒ cos 2A = 2 - 2 sin² A - 1

⇒ cos 2A = 1 - 2 sin² A