1. Prove csc(symbol for angle) tan(symbol for angle) = sec(symbol for angle)
2. Prove cosx+sinxtanx=secx
1.
cscØtanØ = secØ
LS = (1/sinØ)(sinØ/cosØ)
= 1/cosØ
= secØ
= RS
2. cosx + sinxtanx = secx
LS = cosx + sinx(sinx/cosx)
= (cos^2 x + sin^2 x)/cosx
= 1/cosx
= secx
= RS
To prove the given trigonometric identities, we will use the fundamental trigonometric ratios and algebraic manipulations. Let's go through each proof step by step:
1. Proving cscθ * tanθ = secθ:
We start with the left-hand side of the equation: cscθ * tanθ.
Using reciprocal identities, we can express cscθ and tanθ in terms of sine and cosine:
cscθ * tanθ = (1/sinθ) * (sinθ/cosθ).
Next, we simplify by canceling out the common factor of sinθ:
cscθ * tanθ = 1/cosθ = secθ.
Therefore, cscθ * tanθ is indeed equal to secθ.
2. Proving cosx + sinx * tanx = secx:
Let's begin with the left-hand side of the equation: cosx + sinx * tanx.
Using the identity tanx = sinx/cosx, we can rewrite the expression:
cosx + sinx * (sinx/cosx).
Next, we simplify by multiplying sinx with sinx/cosx:
cosx + (sinx^2 / cosx).
To further simplify and combine the terms, we find a common denominator:
(cos^2x + sinx^2) / cosx.
Using the Pythagorean identity cos^2x + sin^2x = 1, we can simplify further:
1 / cosx = secx.
Hence, cosx + sinx * tanx simplifies to secx.
Therefore, the equation cosx + sinxtanx = secx is indeed true.