What is the general solution for sin^2(x)*cos^2(x)=1 ?

The general solution for sin^2(x)*cos^2(x)=1 is No Solution for x\in\mathbb{R}

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Popular >

sin^2(x)*cos^2(x)=1

Solution

sin^{2} (x) \cdot cos^{2} (x) = 1

Solution

No Solution for x \in R

Solution steps

sin^{2} (x) cos^{2} (x) = 1

Subtract

1

from both sides

sin^{2} (x) cos^{2} (x) - 1 = 0

Factor

sin^{2} (x) cos^{2} (x) - 1 : (sin (x) cos (x) + 1) (sin (x) cos (x) - 1)

sin^{2} (x) cos^{2} (x) - 1

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

sin^{2} (x) cos^{2} (x) = (sin (x) cos (x))^{2}

= (sin (x) cos (x))^{2} - 1

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(sin (x) cos (x))^{2} - 1 = (sin (x) cos (x) + 1) (sin (x) cos (x) - 1)

= (sin (x) cos (x) + 1) (sin (x) cos (x) - 1)

(sin (x) cos (x) + 1) (sin (x) cos (x) - 1) = 0

Solving each part separately

sin (x) cos (x) + 1 = 0 or sin (x) cos (x) - 1 = 0

sin (x) cos (x) + 1 = 0 :

No Solution

sin (x) cos (x) + 1 = 0

Rewrite using trig identities

sin (x) cos (x) + 1

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 1 + \frac{sin ( 2 x )}{2}

1 + \frac{sin ( 2 x )}{2} = 0

Move

1

to the right side

1 + \frac{sin ( 2 x )}{2} = 0

Subtract

1

from both sides

1 + \frac{sin ( 2 x )}{2} - 1 = 0 - 1

Simplify

\frac{sin ( 2 x )}{2} = - 1

\frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 2 (- 1)

Simplify

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

sin (x) cos (x) - 1 = 0 :

No Solution

sin (x) cos (x) - 1 = 0

Rewrite using trig identities

sin (x) cos (x) - 1

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 1 + \frac{sin ( 2 x )}{2}

- 1 + \frac{sin ( 2 x )}{2} = 0

Move

1

to the right side

- 1 + \frac{sin ( 2 x )}{2} = 0

Add

1

to both sides

- 1 + \frac{sin ( 2 x )}{2} + 1 = 0 + 1

Simplify

\frac{sin ( 2 x )}{2} = 1

\frac{sin ( 2 x )}{2} = 1

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = 1

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)*cos^2(x)=1 ?
The general solution for sin^2(x)*cos^2(x)=1 is No Solution for x\in\mathbb{R}