Prove that tan (Beta) sin (Beta) + cos (Beta) = sec (Beta)
Please explain.
tan*sin + cos
sin^2/cos + cos^2/cos
(sin^2+cos^2)/cos
1/cos
sec
sec (beta) cot(beta)
To prove that tanaβ sinβ + cosβ = secβ, we will start with the left-hand side of the equation and use trigonometric identities to simplify.
Step 1: Start with tanaβ sinβ + cosβ.
Step 2: Replace tanaβ with sinαβ/cosαβ, where αβ represents the angle between the x-axis and the terminal side of angle β in standard position.
Step 3: tanaβ sinβ + cosβ becomes (sinαβ/cosαβ)sinβ + cosβ.
Step 4: Apply the identity sin2θ + cos2θ = 1 by writing sinαβ as sinαβ = sinβ * cosαβ and cosαβ as cosαβ = cosβ * cosαβ.
Step 5: (sinβ * cosαβ/cosαβ)sinβ + cosβ simplifies to sinβ * sinβ + cosβ.
Step 6: Rewrite sin2β as 1 - cos2β.
Step 7: sinβ * sinβ + cosβ becomes 1 - cos2β + cosβ.
Step 8: Combine like terms. 1 - cos2β + cosβ simplifies to 1 + cosβ - cos2β.
Step 9: Apply the identity 1 + cosθ = secθ and cos2θ = 1 - sin2θ.
Step 10: 1 + cosβ - cos2β becomes secβ.
Hence, tanaβ sinβ + cosβ equals secβ.
To prove the trigonometric identity tan(Beta) sin(Beta) + cos(Beta) = sec(Beta), we'll start with the left-hand side of the equation and manipulate it step by step until we reach the right-hand side.
Step 1: Recall the definitions of the trigonometric functions:
- tan(Beta) = sin(Beta) / cos(Beta)
- sec(Beta) = 1 / cos(Beta)
Step 2: Substitute the definitions into the left-hand side of the equation:
tan(Beta) sin(Beta) + cos(Beta)
= (sin(Beta) / cos(Beta)) sin(Beta) + cos(Beta)
Step 3: Simplify by multiplying (sin(Beta) / cos(Beta)) by sin(Beta):
= (sin^2(Beta) / cos(Beta)) + cos(Beta)
Step 4: To add these two terms, we need a common denominator. The least common denominator is cos(Beta). Multiply the first term (sin^2(Beta) / cos(Beta)) by (cos(Beta) / cos(Beta)):
= (sin^2(Beta) / cos(Beta)) * (cos(Beta) / cos(Beta)) + cos(Beta)
Step 5: Simplify:
= (sin^2(Beta) * cos(Beta)) / (cos(Beta))^2 + cos(Beta)
= sin^2(Beta) / cos(Beta) + cos(Beta)
Step 6: Recall the definition of sec(Beta):
= sec(Beta) + cos(Beta)
= cos(Beta)^-1 + cos(Beta)
Step 7: Combine like terms, using a common denominator:
= (1 + cos^2(Beta)) / cos(Beta)
Step 8: Apply the Pythagorean identity sin^2(Beta) + cos^2(Beta) = 1:
= 1 / cos(Beta)
= sec(Beta)
Thus, we have shown that tan(Beta) sin(Beta) + cos(Beta) is equal to sec(Beta).