Math  /  Algebra

Questionk) 1sin2xcos2xcos4x=tan4x+tan2x+1\frac{1-\sin ^{2} x \cos ^{2} x}{\cos ^{4} x}=\tan ^{4} x+\tan ^{2} x+1

Studdy Solution
Compare both sides to verify the equality:
From Step 1, the left-hand side simplifies to sec2x \sec^2 x .
From Step 2, the right-hand side simplifies to sec4xsec2x+1 \sec^4 x - \sec^2 x + 1 .
To verify the equality, we need to check if:
sec2x=sec4xsec2x+1 \sec^2 x = \sec^4 x - \sec^2 x + 1
Rearrange the equation:
sec4x2sec2x+1=0 \sec^4 x - 2\sec^2 x + 1 = 0
Let y=sec2x y = \sec^2 x , then:
y22y+1=0 y^2 - 2y + 1 = 0
This is a perfect square:
(y1)2=0 (y - 1)^2 = 0
Thus, y=1 y = 1 , which implies sec2x=1 \sec^2 x = 1 , hence cos2x=1 \cos^2 x = 1 .
This is true when x=nπ x = n\pi for integer n n .
Therefore, the original equation holds true under these conditions.
The equation is verified to be true under the condition x=nπ x = n\pi .

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