Prove that: √{2 + √(2 + 2cos 4A)} = 2 cosA | SciPiPupil

Prove that: $\sqrt{2 +\sqrt{2+ 2cos 4A}} = 2 cosA$

This is a class 10 Question From Trigonometric Identities chapter of Unit Trigonometry. All the steps for the solutions are mentioned as hint. If that's hard for you to navigate, you can always visit the facebook link given at the end of every posts. 

Solution:

Taking LHS,

= $\sqrt{2 +\sqrt{2+ 2cos 4A}}$

[Hint: 4A = 2A +2A]

= $\sqrt{2 +\sqrt{2+ 2cos (2A +2A)}}$

[Hint: cos 2A = 2cos²A -1]

= $\sqrt{2 +\sqrt{2+ 2(2cos² 2A -1)}}$

= $\sqrt{2 +\sqrt{2+ 4cos² 2A -2}}$

[Hint: +2-2 = 0]

= $\sqrt{2 +\sqrt{4cos² 2A}}$

[Hint: 4cos² 2A = (2cos2A)(2cos2A) so, √(4cos²2A) = 2cos2A]

= $\sqrt{2 +2cos 2A}$

[Hint: cos2A = 2cos²A -1]

= $\sqrt{2 +2(2cos²A -1)}$

= $\sqrt{2 +4cos²A -2}$

= $\sqrt{4cos²A}$

= $2cos A$

= RHS

Here is the Facebook link to the solution of this question in image. 

Related Notes:

Link: Introduction To Trigonometry

Link: Values of Trigonometric Ratios

Link: Compound Angles

Question: Prove that: √{2 + √(2 + 2cos 4A)} = 2 cosA | SciPiPupil

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