Question

If $\cos x+{{\cos }^{2}}x=1$ , then the value of is ${{\sin }^{12}}x+3{{\sin }^{10}}x+3{{\sin }^{8}}x+si{{n}^{6}}x-1$ is

1. $2$
2. $1$
3. $-1$
4. $0$

Answer 1

In this type of question you need to use the given equations in a way so that it can be of the form of the equation which we need to prove, basically you need to simplify or expand the equations using basic identities of trigonometry and hence you will get your required expression.

Complete step by step answer:
Here we will use the basic trigonometric identities to solve the question, by trigonometric identities what I mean is,
Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Identity inequalities which are true for every value occurring on both sides of an equation. Geometrically, these identities involve certain functions of one or more angles. There are various distinct identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.
As it is given in the question,
$\cos x+{{\cos }^{2}}x=1$
Now if we arrange above equation by simply changing the positions of specific terms we get,
$\Rightarrow \cos x=1-{{\cos }^{2}}x$
Now we will use the trigonometric identity and that is:
${{\sin }^{2}}x+{{\cos }^{2}}x=1$
So after using above identity in above equation we get,
$\Rightarrow \cos x={{\sin }^{2}}x$ $......(1)$
Now again using the equation given in the question we have,
$\cos x+{{\cos }^{2}}x=1$
Now to find the value of the given expression in the question we need to observe that the given equation is cubic, the only way to solve further is to observe, although you can try for any other degree equations too but you will find that the equation which value is to be found is cubic.
On cubing the above equation on both sides,
So we have,
$\Rightarrow {{(\cos x+{{\cos }^{2}}x)}^{3}}={{(1)}^{3}}$
Now we will use the algebraic formula for expanding the above equation and that is:
${{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)$
So after using above formula in above equation we get,
$\Rightarrow {{\cos }^{3}}x+{{\cos }^{6}}x+3(\cos x)({{\cos }^{2}}x)(\cos x+{{\cos }^{2}}x)=1$
Now on further simplifying the equation we get,
$\Rightarrow {{\cos }^{3}}x+{{\cos }^{6}}x+3{{\cos }^{4}}x+3{{\cos }^{5}}x=1$ $......(2)$
Now on substituting equation $(1)$ in $(2)$ we get,
$\Rightarrow {{({{\sin }^{2}}x)}^{3}}+{{({{\sin }^{2}}x)}^{6}}+3{{({{\sin }^{2}}x)}^{4}}+3{{({{\sin }^{2}}x)}^{5}}=1$
Now on expanding the above equation we get,
$\Rightarrow {{\sin }^{12}}x+3{{\sin }^{10}}x+3{{\sin }^{8}}x+si{{n}^{6}}x=1$
Now on moving every term to same side in the above equation we get,
$\Rightarrow {{\sin }^{12}}x+3{{\sin }^{10}}x+3{{\sin }^{8}}x+si{{n}^{6}}x-1=0$
Hence, we get the value of above equation as
${{\sin }^{12}}x+3{{\sin }^{10}}x+3{{\sin }^{8}}x+si{{n}^{6}}x-1=0$

So, the correct answer is “Option 4”.

Note: Trigonometry can be used to roof a house, to make the roof inclined (in the case of single individual bungalows) and the height of the roof in buildings etc. It is used in the naval and aviation industries. It is used in cartography (creation of maps).It contributes in calculus also.