Question

if cosec theta minus sin theta equal to M and sec theta minus cos theta equal to N prove that m square n raised to 2 by 3 + MN raised 2 the whole raised to 2 / 3 equal to 1.​

Answer 1

The given expression cannot be proven to be equal to 1.

To prove the given expression, we need to manipulate the equations using trigonometric identities and simplify them to match the desired form.

Given:

1. cosec(theta) - sin(theta) = M

2. sec(theta) - cos(theta) = N

We will start by manipulating equation 1 using the reciprocal identity:

1. cosec(theta) - sin(theta) = M

(1/sin(theta)) - sin(theta) = M

Next, we square both sides of equation 1 to eliminate the square roots:

3.
`(1/sin(theta))^2` - 2*(1/sin(theta))*sin(theta) +
`sin^2`(theta) =
`M^2`

Now, using the Pythagorean identity
`sin^2`(theta) +
`cos^2`(theta) = 1, we can substitute
`cos^2`(theta) = 1 -
`sin^2`(theta) into equation 3:

4.
`(1/sin(theta))^2` - 2*(1/sin(theta))*sin(theta) + (1 -
`sin^2`(theta)) =
`M^2`

Simplifying equation 4:

5.
`(1/sin(theta))^2` - 2 + 2*sin^2(theta) -
`sin^2`(theta) =
`M^2`

`(1/sin(theta))^2` +
`sin^2`(theta) - 2 =
`M^2`

Using the Pythagorean identity
`cosec^2`(theta) = 1 +
`cot^2`(theta), we can rewrite equation 5:

6. (
`cosec^2`(theta) +
`sin^2`(theta)) - 2 =
`M^2`

(
`cosec^2`(theta) +
`sin^2`(theta)) =
`M^2` + 2

We can repeat the same steps for equation 2:

7.
`(sec(theta) - cos(theta))^2`=
`N^2` + 2

Now, we substitute the left-hand side of equations 6 and 7 into the given expression:

8. (
`cosec^2`(theta) +
`sin^2`(theta))^2*(
`sec^2`(theta) +
`cos^2`(theta))^2 / 3 = 1

Using the Pythagorean identities,
`cosec^2`(theta) = 1 +
`cot^2`(theta) and
`sec^2`(theta) = 1 +
`tan^2`(theta), we can simplify equation 8:

9. (1 +
`cot^2`(theta) +
`sin^2`(theta))^2*(1 +
`tan^2`(theta) +
`cos^2`(theta))^2 / 3 = 1

Finally, using the identity
`cot^2`(theta) + 1 =
`cosec^2`(theta) and
`tan^2`(theta) + 1 =
`sec^2`(theta), we simplify equation 9:

10.
`(cosec^2(theta) + sin^2(theta))^2*(sec^2(theta) + cos^2(theta))^{2 / 3` = 1

Since
`(cosec^2(theta) + sin^2(theta))^2 = 1 and (sec^2(theta) + cos^2(theta))^2` = 1, we can simplify equation 10:

11. 1 * 1 / 3 = 1

1 / 3 = 1

However, this is not a true statement. Therefore, the given expression cannot be proven to be equal to 1.