if sin 7x = cos x+10 find the value of x

To find the value of x in the equation sin 7x = cos x+10, we can use trigonometric identities and algebraic manipulation.

Step 1: Use the identity sin^2(x) + cos^2(x) = 1.
Squaring both sides of the equation sin 7x = cos x+10, we get:
(sin 7x)^2 = (cos x+10)^2

Step 2: Apply the double angle formula for sin(2θ):
(sin θ)^2 = 1/2 * (1 - cos(2θ))

Using this formula, we can rewrite the squared terms:
1/2 * (1 - cos(14x)) = 1/2 * (1 + 2*cos(x) + 100)

Step 3: Simplify the equation:
1 - cos(14x) = 1 + 2*cos(x) + 100
- cos(14x) = 2*cos(x) + 100

Step 4: Rearrange the equation:
cos(14x) = -2*cos(x) - 100

Step 5: Apply the identity cos(θ) = cos(-θ):
cos(14x) = cos(-x)

Since cosine is an even function, the only way for the equation to hold true is if the angle inside the cosine function is either x or -x.

Therefore, we can set up two equations:
14x = x
14x = -x

Solving the first equation:
14x - x = 0
13x = 0
x = 0

Solving the second equation:
14x + x = 0
15x = 0
x = 0

In both cases, x = 0.

Therefore, the value of x that satisfies the given equation is x = 0.