Question

Find the value of cos(x) if sin^2(x) - 1/cos(x) = -1.

a) cos(x) = 0
b) cos(x) = -1
c) cos(x) = 1
d) cos(x) = 2

Answer 1

By applying trigonometric identities, it's shown that the suitable answer for cos(x), given the equation sin^2(x) - 1/cos(x) = -1, is b) cos(x) = -1, as cos(x) = 0 would make the given equation undefined.

Step-by-step explanation:

To find the value of cos(x) given that sin2(x) - 1/cos(x) = -1, we can use trigonometric identities. First, recognize that sin2(x) can be converted to 1 - cos2(x) using the Pythagorean identity: sin2(x) + cos2(x) = 1, which implies sin2(x) = 1 - cos2(x).

Now substitute into the original equation:

• 1 - cos2(x) - 1/cos(x) = -1
• Multiply by cos(x) to clear the denominator:
• cos(x) - cos2(x) - 1 = -cos(x)
• Add cos2(x) to both sides:
• cos(x) - 1 = -cos2(x)
• Now, add cos2(x) + 1 to both sides:
• cos(x) + cos2(x) = 0
• Factor out cos(x):
• cos(x)(1 + cos(x)) = 0

For this product to be zero, one of the factors must be zero:

• cos(x) = 0
• cos(x) = -1

However, the options provided in the question don't include cos(x) = 0 as the value needs to be finite for the original equation to be true since 1/0 is undefined. Thus, the only suitable answer from the options provided is b) cos(x) = -1. Please note, option a) couldn't be the answer as it would make the original equation undefined due to division by zero.