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

Studdy Solution

STEP 1

1. We will simplify the left-hand side of the equation.
2. We will simplify the right-hand side of the equation.
3. We will check if both sides are equal after simplification.

STEP 2

1. Simplify the left-hand side of the equation.
2. Simplify the right-hand side of the equation.
3. Compare both sides to verify the equality.

STEP 3

Simplify the left-hand side of the equation:
The left-hand side is given by:
$\frac{1 - \sin^2 x \cos^2 x}{\cos^4 x}$
We know the identity $\sin^2 x = 1 - \cos^2 x$ . However, it is more useful here to use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ .
First, notice that:
$1 - \sin^2 x \cos^2 x = \cos^4 x + \sin^2 x \cos^2 x$
This can be rewritten using the identity $\sin^2 x + \cos^2 x = 1$ :
$= \cos^4 x + \sin^2 x \cos^2 x = \cos^2 x (\cos^2 x + \sin^2 x) = \cos^2 x$
Thus, the left-hand side becomes:
$\frac{\cos^2 x}{\cos^4 x} = \frac{1}{\cos^2 x} = \sec^2 x$

STEP 4

Simplify the right-hand side of the equation:
The right-hand side is given by:
$\tan^4 x + \tan^2 x + 1$
Recall that $\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$ and $\tan^4 x = \left(\frac{\sin^2 x}{\cos^2 x}\right)^2 = \frac{\sin^4 x}{\cos^4 x}$ .
Using the identity $\tan^2 x = \sec^2 x - 1$ , we can rewrite the expression as:
$(\sec^2 x - 1)^2 + (\sec^2 x - 1) + 1$
Expanding $(\sec^2 x - 1)^2$ :
$= \sec^4 x - 2\sec^2 x + 1$
Thus, the right-hand side becomes:
$\sec^4 x - 2\sec^2 x + 1 + \sec^2 x - 1 + 1$
Simplify:
$= \sec^4 x - \sec^2 x + 1$

STEP 5

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

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms