Can someone check my answers please!!!
Simplify (tan ^2 theta csc^2 theta-1)/(tan^2 theta).
My answer: 1
Simplify ((cos x)/(sec x-1)) + ((cos x) /(sec x +1)
My answer: 2cot^2 x
Find a numerical value of one trigonometric function of x if (tan x/cot x )– (sec x/cos x ) = (2/csc x).
My answer: sin x= -1/2
I agree with all 3 results.
To simplify the expression (tan^2 theta csc^2 theta-1)/(tan^2 theta), follow these steps:
Step 1: Simplify the numerator.
- Expand (tan^2 theta csc^2 theta) to (tan^2 theta * 1/sin^2 theta).
- Multiply the expressions: (tan^2 theta * 1/sin^2 theta) = (tan^2 theta / sin^2 theta).
Step 2: Simplify the denominator.
- The denominator is already in its simplest form: tan^2 theta.
Step 3: Combine the numerator and denominator.
- Divide the numerator (tan^2 theta / sin^2 theta) by the denominator (tan^2 theta):
(tan^2 theta / sin^2 theta) / (tan^2 theta) = 1/sin^2 theta.
Therefore, the simplified expression is 1/sin^2 theta or csc^2 theta.
For the second expression ((cos x)/(sec x-1)) + ((cos x) /(sec x +1)), follow these steps to simplify it:
Step 1: Simplify each term.
- For the first term: (cos x)/(sec x-1) can be rewritten as (cos x)/(1/cos x - 1).
- Simplify the denominator by finding the common denominator: (1/cos x - 1) = ((1 - cos x)/cos x).
- Therefore, the first term simplifies to: (cos x)/(1 - cos x).
- For the second term, (cos x)/(sec x + 1), we can rewrite it as (cos x)/(1/cos x + 1).
- Simplify the denominator: (1/cos x + 1) = ((1 + cos x)/cos x).
- Therefore, the second term simplifies to: (cos x)/(1 + cos x).
Step 2: Combine the terms.
- Add the two simplified terms: (cos x)/(1 - cos x) + (cos x)/(1 + cos x).
Step 3: Find a common denominator and simplify.
- The common denominator of (1 - cos x) and (1 + cos x) is (1 - cos^2 x) = sin^2 x.
- Multiply the first term by (1 + cos x)/(1 + cos x) and the second term by (1 - cos x)/(1 - cos x):
[(cos x)(1 + cos x)]/(sin^2 x) + [(cos x)(1 - cos x)]/(sin^2 x).
- Simplify the numerator: cos x + cos^2 x - cos x + cos^2 x = 2cos^2 x.
- The final simplified expression is: 2cos^2 x/sin^2 x, or 2cot^2 x.
To find a numerical value of one trigonometric function of x if (tan x/cot x )– (sec x/cos x ) = (2/csc x), follow these steps:
Step 1: Simplify each term.
- Rewrite tan x/cot x as sin x/cos x.
- Rewrite sec x/cos x as 1/cos x.
Step 2: Substitute the expressions back into the equation.
- Replace tan x/cot x with sin x/cos x and sec x/cos x with 1/cos x:
(sin x/cos x) - (1/cos x) = 2/csc x.
Step 3: Simplify the equation.
- The left side of the equation now has a common denominator of cos x:
(sin x - 1)/cos x = 2/sin x.
Step 4: Cross-multiply and solve for sin x.
- Cross-multiply the equation: (sin x - 1) * sin x = 2 * cos x.
- Distribute: sin^2 x - sin x = 2 cos x.
- Rearrange the equation: sin^2 x - sin x - 2 cos x = 0.
Step 5: Solve the quadratic equation.
- Factor the quadratic equation: (sin x - 2)(sin x + 1) = 0.
- Set each factor equal to zero: sin x - 2 = 0 or sin x + 1 = 0.
- Solve for sin x: sin x = 2 or sin x = -1.
Therefore, the numerical value of sin x can be -1/2.