Prove: 1-(sin^2(theta)/(1+cos(theta)) = cos(theta)

Answer 1

So two hints here:

1) Knowing that sin^2(θ) + cos^2(θ)=1

2) Converting everything to a common denomiantor.

Full answer:

Left side = 1 - (sin^2(θ))/(1+cos(θ)).

Let's convert 1 into: (1+cos(θ))/(1+cos(θ)) so we have a common denominator.

Thus, we have:

Left side = (1 + cos(θ) - sin^2(θ))/(1+cos((θ))

Now, knowing that sin^2(θ) + cos^2(θ)=1, we can substitute sin^2(θ) + cos^2(θ) for the '1' in the numerator.

This gives us:

Left side = (sin^2(θ) + cos^2(θ) + cos(θ) - sin^2(θ))/(1+cos((θ))

The 'sin^2(θ)' and '-sin^2(θ)' terms in the numerator cancel out, giving us:

Left side = (cos^2(θ) + cos(θ))/(1+cos(θ))

Now, we can factor cos^2(θ) + cos(θ) in the numerator into: cos(θ)*[1+ cos(θ)]

This gives us:

Left side = cos(θ)*[1+ cos(θ)] / (1 + cos(θ))

The (1+cos(θ)) terms in the numerator and denominator cancel out, giving us:

Left side = cos(θ) = Right side