Question

The value of 2$(si{n}^6{735}^{\circ }+co{s}^6{735}^{\circ })$ - 3 $(si{n}^4{735}^{\circ }+co{s}^4{735}^{\circ })$ + 1 is __.

Answer 1

Given expression

2($si{n}^6{735}^{\circ }+co{s}^6{735}^{\circ }$) - 3 ($si{n}^4{735}^{\circ }+co{s}^4{735}^{\circ }$) + 1

Sin (${735}^{\circ }$) = sin (2 $\times$ ${360}^{\circ }$ + ${15}^{\circ }$)

= sin ${15}^{\circ }$

= sin (${45}^{\circ }$-${30}^{\circ }$) = sin${45}^{\circ }$cos${30}^{\circ }$ - cos${45}^{\circ }$ sin${30}^{\circ }$

= $\frac{1}{\sqrt{2}}$ $\times$ $\frac{\sqrt{3}}{2}$ - $\frac{1}{\sqrt{2}}$ $\times$ $\frac{1}{2}$ = $\frac{\sqrt{3}-1}{2\sqrt{2}}$

Similarly cos ${735}^{\circ }$ = cos (2 $\times$ ${360}^{\circ }$ + ${15}^{\circ }$) = cos${15}^{\circ }$

Cos (${45}^{\circ }$-${30}^{\circ }$) = cos${45}^{\circ }$cos${30}^{\circ }$ + sin${45}^{\circ }$.sin${30}^{\circ }$

= $\frac{1}{\sqrt{2}}$ $\times$ $\frac{\sqrt{3}}{2}$ + $\frac{1}{\sqrt{2}}$ $\times$ $\frac{1}{2}$

= $\frac{\sqrt{3}+1}{2\sqrt{2}}$

Substituting sin ${735}^{\circ }$ & cos ${735}^{\circ }$ is the given expression.

$2.[{\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^6+{\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)}^6]-3[{\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^4+{\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)}^4]+1$

Can we further simplify the given expression?

We might be able to simplify. But It's going to be time consuming. It doesn't look like an effective method to solve the given expression.

Just look at the expression it contains the terms of $si{n}^{6}x$ + $co{s}^{6}x$

If we use the identity $a^3$ + $b^3$ = (a+b)($a^2$-ab+$b^3$)

Where a & b are $si{n}^2{735}^{\circ }$ & $co{s}^2{735}^{\circ }$ respectively. Using the identity $si{n}^{2}x$ + $co{s}^{2}x$ = 1 we can easily simplify the expression.

2[$(si{n}^2{735}^{\circ }{)}^3$+ $(co{s}^2{735}^{\circ }{)}^3$]- 3($si{n}^4{735}^{\circ }$ + $co{s}^4{735}^{\circ }$) + 1

$a^3$ + $b^3$ = (a+b) ($a^2$ + $b^2$ - ab)

2[($si{n}^2{735}^{\circ }$+ $co{s}^2{735}^{\circ }$) ($si{n}^4{735}^{\circ }$ + $co{s}^4{735}^{\circ }$ - $si{n}^2{735}^{\circ }$ $co{s}^2{735}^{\circ }$)] - 3($si{n}^4{735}^{\circ }$+ $co{s}^4{735}^{\circ }$) + 1

Using identity $si{n}^{2}x$ + $co{s}^{2}x$ = 1

2($si{n}^4{735}^{\circ }$ + $co{s}^4{735}^{\circ }$)- 2$si{n}^2{735}^{\circ }$ $co{s}^2{735}^{\circ }$ - 3($si{n}^4{735}^{\circ }$ + $co{s}^4{735}^{\circ }$) + 1

= -$si{n}^4{735}^{\circ }$ - $co{s}^4{735}^{\circ }$ - 2$si{n}^2{735}^{\circ }$ $co{s}^2{735}^{\circ }$ + 1

=-[$(si{n}^2{735}^{\circ }{)}^2$+ $(co{s}^2{735}^{\circ }{)}^2$ + 2($si{n}^2{735}^{\circ }$) ($co{s}^2{735}^{\circ }$)]+1

= - $(si{n}^2{735}^{\circ }+co{s}^2{735}^{\circ }{)}^2$+ 1

= -1+1 = 0