1) cscθsinθ-sin^2θ
2)sinθcscθ/tanθ
3)secθ/cotθ+tanθ
4)(1-cosθ)(1+cosθ)
Is there a question ?
how do you solve them
1) To simplify the expression cscθsinθ - sin^2θ, we can first use the identity cscθ = 1/sinθ.
So, the expression becomes (1/sinθ)sinθ - sin^2θ.
Now, we can simplify further by canceling out the sinθ terms, giving us 1 - sin^2θ.
Lastly, we can use the trigonometric identity sin^2θ + cos^2θ = 1.
Substituting this identity into our expression, we get 1 - (1 - cos^2θ).
Simplifying this expression, we have 1 - 1 + cos^2θ, which simplifies to cos^2θ.
Therefore, the simplified expression is cos^2θ.
2) To simplify the expression sinθcscθ/tanθ, we can first use the identity cscθ = 1/sinθ.
So, the expression becomes sinθ * (1/sinθ) / tanθ.
Next, we can cancel out the sinθ terms, giving us 1 / tanθ.
Lastly, we can use the trigonometric identity tanθ = sinθ/cosθ.
Substituting this identity into our expression, we get 1 / (sinθ/cosθ).
To divide by a fraction, we can multiply by the reciprocal of that fraction.
Therefore, our expression simplifies to cosθ.
3) To simplify the expression secθ / (cotθ + tanθ), we can first use the identity secθ = 1/cosθ and cotθ = 1/tanθ.
So, the expression becomes (1/cosθ) / (1/tanθ + tanθ).
Next, we can simplify further by taking the reciprocal of the denominator, which is equal to tanθ + 1/tanθ.
To simplify this, we can use the trigonometric identity tanθ + 1/tanθ = secθ + cosecθ.
Therefore, our expression simplifies to (1/cosθ) / (secθ + cosecθ).
To divide by a fraction, we can multiply by the reciprocal of that fraction.
So, our expression becomes (1/cosθ) * (1/(secθ + cosecθ)).
Multiplying the numerators and the denominators, we get 1 / (cosθ * (secθ + cosecθ)).
Further simplification isn't possible unless we have additional information about the values of the trigonometric functions.
4) To simplify the expression (1 - cosθ)(1 + cosθ), we can use the identity (a - b)(a + b) = a^2 - b^2.
Applying this identity to our expression, we have (1^2 - cos^2θ).
Since 1^2 = 1, our expression simplifies to 1 - cos^2θ.