6.Prove that tan y cos^2 y + sin^2y/sin y = cos y + sin �y
10.Prove that 1+tanθ/1-tanθ = sec^2θ+2tanθ/1-tan^2θ
17.Prove that sin^2w-cos^2w/tan w sin w + cos w tan w = cos w-cot w cos w
23.Find a counterexample to shows that the equation sec a� – cos a� = sin a� sec a� is not an identity.
you gotta add some parentheses, so we can tell what's the numerator and what's the denominator.
For #23, just pick an easy value, like pi/4:
√2 - 1/√2 =?= 1/√2 * √2
No.
To prove the given equations, we will simplify both sides of the equation separately and then show that they are equal. Let's start with each one.
6. Prove that tan(y)cos^2(y) + sin^2(y)/sin(y) = cos(y) + sin(y)
First, simplify the left side of the equation:
tan(y)cos^2(y) + sin^2(y)/sin(y)
= sin(y)/cos(y) * cos^2(y) + sin^2(y)/sin(y)
= sin(y) * cos(y) + sin(y)
= sin(y) * (cos(y) + 1)
Now, let's simplify the right side of the equation:
cos(y) + sin(y)
Since we have arrived at the same expression on both sides, we have proved the equation:
tan(y)cos^2(y) + sin^2(y)/sin(y) = cos(y) + sin(y)
10. Prove that (1 + tanθ) / (1 - tanθ) = sec^2θ + 2tanθ / (1 - tan^2θ)
First, simplify the left side of the equation:
(1 + tanθ) / (1 - tanθ)
= (1 + sinθ / cosθ) / (1 - sinθ / cosθ)
= [(cosθ + sinθ) / cosθ] / [(cosθ - sinθ) / cosθ]
= (cosθ + sinθ) / (cosθ - sinθ)
Now, let's simplify the right side of the equation:
sec^2θ + 2tanθ / (1 - tan^2θ)
= (1 / cos^2θ) + (2sinθ / cosθ) / (1 - sin^2θ / cos^2θ)
= (1 + 2sinθcosθ) / (cos^2θ - sin^2θ)
= (cosθ + sinθ) / (cosθ - sinθ)
Since we have arrived at the same expression on both sides, we have proved the equation:
(1 + tanθ) / (1 - tanθ) = sec^2θ + 2tanθ / (1 - tan^2θ)
17. Prove that (sin^2w - cos^2w) / (tanw sinw + cosw tanw) = cosw - cotw cosw
First, simplify the left side of the equation:
(sin^2w - cos^2w) / (tanw sinw + cosw tanw)
= [(sinw)^2 - (cosw)^2] / (sinw(tanw + cosw))
= [(sinw - cosw)(sinw + cosw)] / (sinw(tanw + cosw))
Now, let's simplify the right side of the equation:
cosw - cotw cosw
= cosw - (cosw / sinw)cosw
= cosw - cos^2w/sinw
= [(sinw cosw) - (cos^2w)] / sinw
= [(sinw - cosw)(cosw)] / sinw
= [(sinw - cosw)(sinw + cosw)] / (sinw(tanw + cosw))
Since we have arrived at the same expression on both sides, we have proved the equation:
(sin^2w - cos^2w) / (tanw sinw + cosw tanw) = cosw - cotw cosw
23. To find a counterexample, we need to find values for α where the equation sec(α) - cos(α) = sin(α) sec(α) is not true.
Let's start by simplifying the equation:
sec(α) - cos(α) = sin(α) sec(α)
1/cos(α) - cos(α) = sin(α)/cos(α)
(1 - cos^2(α))/cos(α) = sin(α)/cos(α)
1 - cos^2(α) = sin(α)
Now, if we choose α = 0, the left side of the equation becomes 1 - cos^2(0) = 1 - 1 = 0.
However, the right side of the equation remains sin(0) = 0.
Since the left and right sides of the equation do not match for α = 0, we have found a counterexample to show that the equation is not an identity.
Therefore, the equation sec(α) - cos(α) = sin(α) sec(α) is not true for all values of α.