Answer:Sin2 theta sec theta = 2 sin theta
Starting with the left-hand side, we can use the identity for sin(2θ):
sin(2θ) = 2sin(θ)cos(θ)
Then, we can substitute this into the left-hand side:
sin(2θ) sec(θ) = 2sin(θ)cos(θ) / cos(θ)
Simplifying the right-hand side, we get:
2sin(θ)
Canceling out the common factor of cos(θ), we get:
sin(2θ) sec(θ) = 2sin(θ)
Therefore, sin2 theta sec theta= 2 sin theta is true.
Csc (θ + 2π) = csc θ
We can use the identity for the cosecant function:
csc(θ) = 1/sin(θ)
Then, we can substitute θ + 2π for θ in the right-hand side:
csc(θ + 2π) = 1/sin(θ + 2π)
Using the periodicity property of the sine function, we know that sin(θ + 2π) = sin(θ). Therefore, we can substitute sin(θ) for sin(θ + 2π):
csc(θ + 2π) = 1/sin(θ)
Which is equivalent to csc θ.
Therefore, Csc (θ + 2π) = csc θ is true.
Tan (θ - π/2) = -cot θ
We can use the identity for the tangent and cotangent functions:
tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)
Substituting θ - π/2 for θ in the left-hand side of the equation:
tan(θ - π/2) = sin(θ - π/2)/cos(θ - π/2)
Using the trigonometric identities for sine and cosine, we get:
sin(θ - π/2) = cos(π/2 - θ)
cos(θ - π/2) = sin(π/2 - θ)
Substituting these identities into the equation, we get:
tan(θ - π/2) = cos(π/2 - θ) / sin(π/2 - θ)
Using the identity for the cotangent function, we can simplify the right-hand side:
cos(π/2 - θ) / sin(π/2 - θ) = cot(π/2 - θ) = cot(θ)
Multiplying the numerator and denominator of the left-hand side by -1, we get:
tan(θ - π/2) = -cos(θ) / sin(θ) = -cot(θ)
Therefore, Tan (θ - π/2) = -cot θ is true.
Cot (θ + π/2) = -tan θ
We can use the identity for the cotangent and tangent functions:
cot(θ) = cos(θ)/sin(θ)
tan(θ) = sin(θ)/cos(θ)
Substituting θ + π/2 for θ in the left-hand side of the equation:
cot(θ + π/2) = cos(θ + π/2) / sin(θ + π/2)
Using the trigonometric identities for sine and cosine, we get:
cos(θ + π/2) = -sin(θ)
sin(θ + π/2) = cos(θ)
Substituting these identities
Explanation: