How do I do these questions? I'm supposed to prove them so that both sides are the same and I'm only supposed to manipulate one side and leave the other side alone but I'm stuck.
Problem 1. (secx/cosx)-(cscx/sinx)=tan^2x-cot^2x
Problem 2. secx+cscx=(1+tanx)/sinx
1.
LS = (1/cosx) / cosx - (1/sinx) /sinx
= 1/cos^2 x - 1/(sin^2 x)
= (sin^2x + cos^2x)/cos^2x - (sin^2x + cos^2x)/sin^2x
= sin^2x/cos^2x + cos^2x/cos^2x - sin^2x/sin^2x - cos^2x/sin^2x
= tan^2x + 1 - 1 - cot^2x
= tan^2x - cot^2x
= RS
2. RS = (1+tanx)/sinx
= 1/sinx + tanx/sinx
= cscx + (sinx/cosx) /sinx
= cscx + 1/cosx
= cscx + secx
= LS
To solve these problems, we will follow a step-by-step approach known as manipulating equations. Here's how you can solve each problem while only manipulating one side and leaving the other side untouched:
Problem 1: `(secx/cosx) - (cscx/sinx) = tan^2x - cot^2x`
Step 1: Start with the more complex side (the left side in this case) and simplify it as much as possible, using trigonometric identities.
Recall the following identities:
- `secx = 1/cosx`
- `cscx = 1/sinx`
- `tanx = sinx/cosx`
- `cotx = cosx/sinx`
Using these identities, we can rewrite the left side of the equation:
`= (1/cosx)/(cosx) - (1/sinx)/(sinx)`
`= 1/cosx * 1/cosx - 1/sinx * 1/sinx`
`= 1/cos^2x - 1/sin^2x`
`= (sin^2x - cos^2x)/(sin^2x * cos^2x)`
Step 2: Simplify the right side of the equation:
`= tan^2x - cot^2x`
`= (sin^2x/cos^2x) - (cos^2x/sin^2x)`
`= (sin^2x - cos^2x)/(sin^2x * cos^2x)`
Step 3: Compare both sides of the equation:
By simplifying both sides, we end up with the same expression: `(sin^2x - cos^2x)/(sin^2x * cos^2x)`. This means that both sides are equal, and the given equation is proven.
Problem 2: `secx + cscx = (1 + tanx) / sinx`
Step 1: Start with the more complex side (the left side in this case) and simplify it as much as possible, using trigonometric identities.
Recall the following identities:
- `secx = 1/cosx`
- `cscx = 1/sinx`
- `tanx = sinx/cosx`
Using these identities, we can rewrite the left side of the equation:
`= 1/cosx + 1/sinx`
Step 2: Simplify the right side of the equation:
`= (1 + tanx) / sinx`
Step 3: Compare both sides of the equation:
By simplifying both sides, we end up with the same expression: `(1 + tanx) / sinx`. This means that both sides are equal, and the given equation is proven.
Remember, when manipulating equations, it's important to use trigonometric identities and properties to simplify each side. Comparing both sides at the end will determine if they are equal or not, proving the equation is true.