Subtract and simplify: (sin theta-1/ cos theata) -( cos theta/ sin theta-1)
do you mean
(sinT - 1/cosT)
or do you mean
(sinT-1)/cosT
(SinT-1)/cosT
To subtract and simplify the given expression, (sin theta - 1/cos theta) - (cos theta/sin theta - 1), we need to use some trigonometric identities and some basic algebraic simplification.
Step 1: Simplify the individual fractions.
- In the first fraction, sin theta - 1/cos theta, we can combine the terms by multiplying the numerator and denominator of the second fraction by cos theta. This gives us: (sin theta * cos theta - 1) / cos theta.
- In the second fraction, cos theta/sin theta - 1, we can simplify by multiplying the numerator and denominator of the second fraction by sin theta. This gives us: (cos theta - sin theta) / (sin theta * sin theta - sin theta).
Step 2: Simplify the denominators.
- In the first fraction, the denominator is already simplified.
- In the second fraction, we can factor out sin theta from the denominator: sin theta * (sin theta - 1).
Step 3: Combine the fractions into a single fraction.
- Now that we have the fractions with the same denominator, we can combine them by subtracting the numerators: [(sin theta * cos theta - 1) - (cos theta - sin theta)] / (sin theta * (sin theta - 1)).
Step 4: Simplify the numerator.
- Expanding the brackets in the numerator gives: sin theta * cos theta - 1 - cos theta + sin theta.
- Combining like terms gives us: (sin theta * cos theta - cos theta) + (sin theta - 1).
Step 5: Factor out common terms.
- In the numerator, we can factor out cos theta from the first two terms and factor out 1 from the last two terms: cos theta * (sin theta - 1) + 1 * (sin theta - 1).
Step 6: Combine like terms in the numerator.
- Now that we have the same factors, we can add them together: (cos theta + 1) * (sin theta - 1).
So, the simplified expression is (cos theta + 1) * (sin theta - 1).