Question

sin A + sin B + sin(A + B)
sin A + sin B - sin(A + B)​

Answer 1

$\color{red} {{{\Large {\bf{To\:\:Simplify\::\frac{\sin ^4(+)-\cos ^4(+)}{\sin ^2(+)-\cos ^2(+)}}}}}}$

$\color{green}{{{\large {\bf{Your\:\:Answer\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}=1}}}}}$

$\color{yellow} {\Huge {\sf{Solution:}}}$

$\color{blue} {\large {\bf{Factor\:\sin ^4(x)-\cos ^4(x)}}}$

$\tt \color{blue} {\mathrm{Rewrite\:}\sin ^4(x)-\cos ^4(x)\mathrm{\:as\:}(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x))^2-(\cos ^2(x))^2}$

$\color{fuchsia} {\normalsize {\mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}}$

$\color{fuchsia} {\normalsize \sin ^4(x)=(\sin ^2(x))^2}$

$\color{fuchsia} {\normalsize =(\sin ^2(x))^2-\cos ^4(x)}$=

$\color{fuchsia} {\normalsize \mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}$

$\color{fuchsia} {\normalsize \cos ^4(x)=(\cos ^2(x))^2}$

$\color{fuchsia} {\normalsize =(\sin ^2(x))^2-(\cos ^2(x))^2}$=

$\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}$x^2-y^2=(x+y)(x-y)

$(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))$

=$(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))$=

$\color{blue} {\large {\bf{Factor\:\sin ^2(x)-\cos ^2(x)}}}$

$\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}$x^2-y^2=(x+y)(x-y)

$\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)-\cos (x))$

(x)=(sin(x)+cos(x))(sin(x)−cos(x))

=(\sin (x)+\cos (x))(\sin (x)-\cos (x))=(sin(x)+cos(x))(sin(x)−cos(x))

$\large=(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))\ \textless \ br /\ \textgreater \ (x))(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ \large =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{\sin ^2(x)-\cos ^2(x)}$=

$\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}$x^2-y^2=(x+y)(x-y)

$\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)$

$(x)=(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}$=

$\mathrm{Cancel\:}\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}:\quad \sin ^2(x)+\cos ^2(x)Cancel \ \textless \ br /\ \textgreater \ (sin(x)+cos(x))(sin(x)−cos(x))$

$\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)+\cos(x)Cancelthecommonfactor:sin(x)+cos(x)$

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

Answer 2

Answer:

ToSimplify:

sin

2

(+)−cos

2

(+)

sin

4

(+)−cos

4

(+)

\color{green}{{{\large {\bf{Your\:\:Answer\::\frac{\sin ^4(x)-\cos ^4(x)}{\sin ^2(x)-\cos ^2(x)}=1}}}}}YourAnswer:

sin

2

(x)−cos

2

(x)

sin

4

(x)−cos

4

(x)

=1

\color{yellow} {\Huge {\sf{Solution:}}}Solution:

\color{blue} {\large {\bf{Factor\:\sin ^4(x)-\cos ^4(x)}}}Factorsin

4

(x)−cos

4

(x)

\tt \color{blue} {\mathrm{Rewrite\:}\sin ^4(x)-\cos ^4(x)\mathrm{\:as\:}(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x))^2-(\cos ^2(x))^2}Rewritesin

4

(x)−cos

4

(x)as(sin

2

(x))

2

−(cos

2

(x))

2

=(sin

2

(x))

2

−(cos

2

(x))

2

\color{fuchsia} {\normalsize {\mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}}Applyexponentrule:a

bc

=(a

b

)

c

\color{fuchsia} {\normalsize \sin ^4(x)=(\sin ^2(x))^2}sin

4

(x)=(sin

2

(x))

2

\color{fuchsia} {\normalsize =(\sin ^2(x))^2-\cos ^4(x)}=(sin

2

(x))

2

−cos

4

(x) =

\color{fuchsia} {\normalsize \mathrm{Apply\:exponent\:rule}:\quad \:a^{bc}=(a^b)^c}Applyexponentrule:a

bc

=(a

b

)

c

\color{fuchsia} {\normalsize \cos ^4(x)=(\cos ^2(x))^2}cos

4

(x)=(cos

2

(x))

2

\color{fuchsia} {\normalsize =(\sin ^2(x))^2-(\cos ^2(x))^2}=(sin

2

(x))

2

−(cos

2

(x))

2

=

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}ApplyDifferenceofTwoSquaresFormula: x^2-y^2=(x+y)(x-y)

(\sin ^2(x))^2-(\cos ^2(x))^2=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))(sin

2

(x))

2

−(cos

2

(x))

2

=(sin

2

(x)+cos

2

(x))(sin

2

(x)−cos

2

(x))

=(\sin ^2(x)+\cos ^2(x))(\sin ^2(x)-\cos ^2(x))(sin

2

(x)+cos

2

(x))(sin

2

(x)−cos

2

(x)) =

\color{blue} {\large {\bf{Factor\:\sin ^2(x)-\cos ^2(x)}}}Factorsin

2

(x)−cos

2

(x)

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}ApplyDifferenceofTwoSquaresFormula: x^2-y^2=(x+y)(x-y)

\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)-\cos (x))sin

2

(x)−cos

2

(x)=(sin(x)+cos(x))(sin(x)−cos(x))

(x)=(sin(x)+cos(x))(sin(x)−cos(x))

=(\sin (x)+\cos (x))(\sin (x)-\cos (x))=(sin(x)+cos(x))(sin(x)−cos(x))

\large=(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))\ \textless \ br /\ \textgreater \ (x))(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ \large =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{\sin ^2(x)-\cos ^2(x)}=(sin

2

(x)+cos

2

(x))(sin(x)+cos(x))(sin(x)−cos(x)) \textless br/ \textgreater (x))(sin(x)+cos(x))(sin(x)−cos(x)) \textless br/ \textgreater =

sin

2

(x)−cos

2

(x)

(sin

2

(x)+cos

2

(x))(sin(x)+cos(x))(sin(x)−cos(x))

=

\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}ApplyDifferenceofTwoSquaresFormula: x^2-y^2=(x+y)(x-y)

\sin ^2(x)-\cos ^2(x)=(\sin (x)+\cos (x))(\sin (x)sin

2

(x)−cos

2

(x)=(sin(x)+cos(x))(sin(x)

(x)=(sin(x)+cos(x))(sin(x)−cos(x))\ \textless \ br /\ \textgreater \ =\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}(x)=(sin(x)+cos(x))(sin(x)−cos(x)) \textless br/ \textgreater =

(sin(x)+cos(x))(sin(x)−cos(x))

(sin

2

(x)+cos

2

(x))(sin(x)+cos(x))(sin(x)−cos(x))

=

\mathrm{Cancel\:}\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)+\cos (x))(\sin (x)-\cos (x))}{(\sin (x)+\cos (x))(\sin (x)-\cos (x))}:\quad \sin ^2(x)+\cos ^2(x)Cancel \ \textless \ br /\ \textgreater \ (sin(x)+cos(x))(sin(x)−cos(x))Cancel

(sin(x)+cos(x))(sin(x)−cos(x))

(sin

2

(x)+cos

2

(x))(sin(x)+cos(x))(sin(x)−cos(x))

:sin

2

(x)+cos

2

(x)Cancel \textless br/ \textgreater (sin(x)+cos(x))(sin(x)−cos(x))

\mathrm{Cancel\:the\:common\:factor:}\:\sin (x)+\cos(x)Cancelthecommonfactor:sin(x)+cos(x)Cancelthecommonfactor:sin(x)+cos(x)Cancelthecommonfactor:sin(x)+cos(x)

=\frac{(\sin ^2(x)+\cos ^2(x))(\sin (x)-\cos (x))}{\sin (x)-\cos (x)}

sin(x)−cos(x)

(sin

2

(x)+cos

2

(x))(sin(x)−cos(x))

=

Step-by-step explanation:

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