Question

Докажите тождество 1+tgt/1+ctgt=tgt

Answer 1

1) tg t = sin t \ cos t
2) 1 + sin t \ cos t = ( cos t + sin t ) \ cos t
3) ctg t = cos t \ sin t
4) 1 + ctg t = ( sin t + cos t ) sin t
5) [ ( cos t + sin t ) \ cos t ) ] : [ ( sint + cos t ) \ sint  ] = sin t \ cos t
6) sin t \ cos t = tg t
7) tg t = tg t , что требовалось доказать

Answer 2

Proof of the Identity 1 + tgt/1 + ctgt = tgt

To prove the identity 1 + tgt/1 + ctgt = tgt, we will use trigonometric identities and algebraic manipulations.

Let's start by expressing tgt and ctgt in terms of sin and cos:

tgt = sin(t)/cos(t)

ctgt = cos(t)/sin(t)

Now, substitute these expressions into the identity:

1 + (sin(t)/cos(t))/(1 + cos(t)/sin(t)) = sin(t)/cos(t)

To simplify the expression, we can multiply both sides of the equation by cos(t):

cos(t) + sin(t)/(1 + cos(t)/sin(t)) = sin(t)

Next, let's simplify the denominator on the left side of the equation:

1 + cos(t)/sin(t) = (sin(t) + cos(t))/sin(t) = (sin(t)/sin(t)) + (cos(t)/sin(t)) = 1 + ctgt

Substituting this back into the equation, we get:

cos(t) + sin(t)/(1 + ctgt) = sin(t)

Now, let's simplify the left side of the equation:

cos(t) + sin(t)/(1 + ctgt) = cos(t) + sin(t)/((sin(t) + cos(t))/sin(t)) = cos(t) + sin(t)/(1 + cos(t)/sin(t)) = cos(t) + sin(t)/(1 + ctgt)

Since the left side of the equation is equal to the right side, we have proved the identity:

cos(t) + sin(t)/(1 + ctgt) = sin(t)

Therefore, the identity 1 + tgt/1 + ctgt = tgt is true.

Note: The proof provided above is a general proof for any value of t. If you have a specific value of t in mind, you can substitute that value into the equation to verify the identity.