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Evaluate $\sin^4{x}$ $+$ $2\sin^2{x}\bigg(1-\dfrac{1}{\csc^2{x}}\bigg)$ $+$ $\cos^4{x}$

A trigonometric expression is defined in terms of three trigonometric functions sine, cosine and cosecant in terms of angle x in this trigonometric problem.

$\sin^4{x}$ $+$ $2\sin^2{x}\bigg(1-\dfrac{1}{\csc^2{x}}\bigg)$ $+$ $\cos^4{x}$

The sine raised to the power of $4$ of an angle $x$ is added to the product of two times sine squared of angle $x$ and the subtraction of the reciprocal of cosecant square of $x$ from $1$. The sum of them is added to the cosine raise to the power of $4$ of an angle $x$. the sum of the three trigonometric expressions has to evaluate by simplification in this trigonometric problem.

Simplify the expression by multiplication

In the second term of the expression, two trigonometric functions are multiplied. The second factor in the second term is a difference of two terms. The difference of them is multiplied by a trigonometric function. The product of them can be evaluated by the distributive property of multiplication over subtraction.

$=\,\,\,$ $\sin^4{x}$ $+$ $2\sin^2{x} \times 1$ $-$ $2\sin^2{x} \times \dfrac{1}{\csc^2{x}}$ $+$ $\cos^4{x}$

Simplify the expression by identities

The reciprocal of cosecant squared of angle $x$ can be converted in sine function as per the reciprocal identity of cosecant.

$=\,\,\,$ $\sin^4{x}$ $+$ $2\sin^2{x}$ $-$ $2\sin^2{x} \times \sin^2{x}$ $+$ $\cos^4{x}$

Now, concentrate on simplifying the trigonometric expression.

$=\,\,\,$ $\sin^4{x}$ $+$ $2\sin^2{x}$ $-$ $2\sin^4{x}$ $+$ $\cos^4{x}$

$=\,\,\,$ $\sin^4{x}$ $-$ $2\sin^4{x}$ $+$ $2\sin^2{x}$ $+$ $\cos^4{x}$

$=\,\,\,$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$ $+$ $\cos^4{x}$

$=\,\,\,$ $\cos^4{x}$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

Express the cosine raised to the power of $4$ of an angle $x$ as the square of cosine square of angle $x$.

$=\,\,\,$ $\big(\cos^2{x}\big)^2$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

According to the cosine squared identity, the square of cosine can be written in terms of square of sine.

$=\,\,\,$ $\big(1-\sin^2{x}\big)^2$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

The expression in the first term represents the square of difference of two terms and it can be expanded as per square of difference formula.

$=\,\,\,$ $1^2$ $+$ $\big(\sin^2{x}\big)^2$ $-$ $2 \times 1 \times \sin^2{x}$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

Evaluate the expression by simplification

Write all terms in a convenient order for simplifying the trigonometric expression and it is helpful for calculating the value of given trigonometric expression.

$=\,\,\,$ $1$ $+$ $\sin^4{x}$ $-$ $2\sin^2{x}$ $-$ $\sin^4{x}$ $+$ $2\sin^2{x}$

$=\,\,\,$ $1$ $+$ $\sin^4{x}$ $-$ $\sin^4{x}$ $-$ $2\sin^2{x}$ $+$ $2\sin^2{x}$

$=\,\,\,$ $1$ $+$ $\cancel{\sin^4{x}}$ $-$ $\cancel{\sin^4{x}}$ $-$ $\cancel{2\sin^2{x}}$ $+$ $\cancel{2\sin^2{x}}$

$=\,\,\,$ $1$

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