Question

please do it .,....​

Answer 1

Required Answer:-

Given:

$\rm \mapsto \sin(x) + { \sin}^{2} (x) = 1$

To find:

$\rm \cos^{12} (x) + 3 { \cos}^{10} (x) + 3 { \cos}^{8}(x) + { \cos}^{6} (x) + 2 { \cos }^{4}(x) + 2 { \cos}^{2}(x) - 2 =?$

Solution:

Given that,

$\rm\sin(x) + { \sin}^{2} (x) = 1$

$\rm \implies\sin(x) = 1 - { \sin}^{2} (x)$

We know that,

$\rm { \cos }^{2}(x) = 1 - { \sin}^{2} (x)$

Therefore,

$\rm \implies\sin(x) = { \cos}^{2} (x)$

Let us assume that,

$\rm \implies { \cos}^{2}(x) = y$

Therefore,

$\rm \cos^{12} (x) + 3 { \cos}^{10} (x) + 3 { \cos}^{8}(x) + { \cos}^{6} (x) + 2 { \cos }^{4}(x) + 2 { \cos}^{2}(x) - 2$

$\rm = {y}^{6} + 3 {y}^{5} + 3 {y}^{4} + {y}^{3} + 2{y}^{2} + 2y - 2$

$\rm = ({y}^{2})^{3} +( 3 \times ({y}^{2})^{2} \times y) +( 3 \times {y}^{2} \times {y}^{2} ) + {y}^{3} + 2{y}^{2} + 2y - 2$

$\rm = ({y}^{2} +y)^{3} + 2{y}^{2} + 2y - 2$

Putting the value of y, we get,

$\rm = ({ \cos}^{4}(x) + { \cos }^{2}(x))^{3} + 2{ \cos}^{4}(x) + 2 { \cos}^{2}(x) - 2$

As cos²(x) = sin(x), So, cos⁴(x) = sin²(x). Thus,

$\rm = ( { \sin }^{2} (x) + { \cos }^{2}(x))^{3} + 2{ \sin}^{2}(x) + 2 { \cos}^{2}(x) - 2$

As we know that, sin²(x) + cos²(x) = 1, So,

$\rm = {1}^{3} + 2 \times 1 - 2$

$\rm = 1 + 2 - 2$

$\rm = 1$

Hence, the Required Answer is 1.

Answer:

• Option B is the answer for the question.

Answer 2

Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, being larger than only Mercury. In English, Mars carries the name of the Roman god of war and is often referred to as the "Red Planet".