Please show how to do these problems so i understand.

1)Prove that sin^2 x (1+cot^2 x)=1

2)Prove that tan x (cot x+tan x)=sec^2 x

well,

1+cot^2 = csc^2

tan(cot+tan) = 1 + tan^2

Sure! Let's go through each problem step by step to understand how to prove them.

1) Proving sin^2 x (1+cot^2 x)=1:
To prove this equation, we will start with the left-hand side of the equation and try to manipulate it until it becomes equal to the right-hand side.

Starting with the left-hand side: sin^2 x (1+cot^2 x)

We know that cot^2 x = 1/tan^2 x, so we can substitute that into the expression: sin^2 x (1+ 1/tan^2 x)

Now, we can find a common denominator for the terms in the parentheses: sin^2 x (tan^2 x/tan^2 x + 1/tan^2 x)

Simplifying the expression inside the parentheses: sin^2 x ((tan^2 x + 1)/tan^2 x)

Using the identity tan^2 x + 1 = sec^2 x, we can further simplify: sin^2 x (sec^2 x/tan^2 x)

Now, we have sin^2 x multiplied by (sec^2 x/tan^2 x). Since sin^2 x = 1 - cos^2 x and tan^2 x = sin^2 x / cos^2 x, we can substitute these identities into the expression: (1 - cos^2 x) (sec^2 x / (sin^2 x / cos^2 x))

Simplifying further: (1 - cos^2 x) (cos^2 x / sin^2 x)

Canceling out common terms: (1 - cos^2 x) / sin^2 x

Using the identity 1 - cos^2 x = sin^2 x, we arrive at: sin^2 x / sin^2 x

Finally, sin^2 x / sin^2 x simplifies to 1.

Hence, we have proved that sin^2 x (1+cot^2 x) equals 1.

2) Proving tan x (cot x + tan x) = sec^2 x:
To prove this equation, we will again start with the left-hand side and manipulate it until it becomes equal to the right-hand side.

Starting with the left-hand side: tan x (cot x + tan x)

We know that cot x = 1/tan x, so we can substitute that into the expression: tan x (1/tan x + tan x)

Now, we want to find a common denominator for the terms in the parentheses: tan x ((1 + tan^2 x)/tan x)

Using the identity 1 + tan^2 x = sec^2 x, we can simplify: tan x (sec^2 x / tan x)

Canceling out common terms: sec^2 x

Hence, we have proved that tan x (cot x + tan x) equals sec^2 x.

By following these step-by-step processes, we can solve these problems and understand how each expression is manipulated to obtain the desired result.