Question

Sin3xcosx-cosxsinx=1

Answer 1

$sin3xcosx-cosxsinx=1\Leftrightarrow \frac{1}{2}sin4x=1\\sin4x=2\Rightarrow x\in \varnothing$

Answer 2

To solve the equation sin(3x)cos(x) - cos(x)sin(x) = 1, we can simplify the expression on the left-hand side using trigonometric identities.

First, let's rewrite the left-hand side using the angle addition formula for sine and cosine:

sin(3x)cos(x) - cos(x)sin(x) = sin(3x + x).

Now, using the angle addition formula, we have:

sin(3x + x) = sin(4x).

So, the equation simplifies to:

sin(4x) = 1.

To solve this equation, we need to find the values of x that satisfy sin(4x) = 1.

The range of the sine function is -1 to 1. The only value of x that gives sin(4x) = 1 is when 4x = π/2 + 2kπ, where k is an integer.

So, we have:

4x = π/2 + 2kπ,

where k is an integer.

Now, we can solve for x:

x = (π/2 + 2kπ)/4,

where k is an integer.

Therefore, the solution for the equation sin(3x)cos(x) - cos(x)sin(x) = 1 is:

x = (π/2 + 2kπ)/4,

where k is an integer.