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Prove that: sec x = 2 / √[2 + √ (2 + 2 cos 4x)]

Trigonometry Problem: Prove the Identity

Question:
Prove that:
sec x = 2 √2 + √2 + 2 cos 4x

Solution / Proof:

We will solve the R.H.S. (Right Hand Side) and prove that it is equal to sec x.

Step 1: Simplify the innermost square root

First, look at the term inside the inner root: (2 + 2 cos 4x).
We can take '2' as common:

= 2 (1 + cos 4x)

Formula Used: 1 + cos 2θ = 2 cos² θ
So, for 4x, it becomes:

= 2 (2 cos² 2x)
= 4 cos² 2x

Now, putting this back into the square root:

√(4 cos² 2x) = 2 cos 2x

Step 2: Update the expression

Now our main expression becomes:

= 2 √2 + 2 cos 2x

Step 3: Simplify the remaining square root

Again, take '2' as common in the denominator:

= 2 (1 + cos 2x)

Using the same formula (1 + cos 2x = 2 cos² x):

= 2 (2 cos² x)
= 4 cos² x

Now, remove the square root:

√(4 cos² x) = 2 cos x

Step 4: Final Answer

Now the expression is:

= 2 2 cos x

Cancel '2' from numerator and denominator:

= 1 / cos x
= sec x

Result: L.H.S. = R.H.S. (Proved)

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