Question

ii) sin^2 (π/4-x) + sin^2( π/4+ x) = 1
( please give answer step by step )​

Answer 1

Answer:

Step-by-step explanation:

Metamorphic rocks were once igneous or sedimentary rocks, but have been changed (metamorphosed) as a result of intense heat and/or pressure within the Earth's crust. They are crystalline and often have a “squashed” (foliated or banded) texture.

Answer 2

$\large\underline{\sf{Solution-}}$

Consider LHS

$\rm :\longmapsto\: {sin}^{2}\bigg[\dfrac{\pi}{4} - x\bigg] + {sin}^{2}\bigg[\dfrac{\pi}{4} + x\bigg]$

On multiply and divide by 2, we get

$\rm \: = \: \dfrac{1}{2}\bigg(2 {sin}^{2}\bigg[\dfrac{\pi}{4} - x\bigg] + 2 {sin}^{2}\bigg[\dfrac{\pi}{4} + x\bigg]\bigg)$

We know,

$\red{ \boxed{ \sf{ \: {2sin}^{2}x = 1 - cos2x}}} \\ \\ \red{ \boxed{ \sf{ \: {2cos}^{2}x = 1 + cos2x}}}$

So, using these, Identities, we get

$\rm \: = \: \dfrac{1}{2}\bigg(1 - cos\bigg[\dfrac{\pi}{2} - 2x\bigg] + 1 - cos\bigg[\dfrac{\pi}{2} + 2 x\bigg]\bigg)$

Now, we know that,

$\red{ \boxed{ \sf{ \:cos\bigg[\dfrac{\pi}{2} - x\bigg] \: = \: sinx}}} \\ \\ \red{ \boxed{ \sf{ \:cos\bigg[\dfrac{\pi}{2} + x\bigg] \: = \: - \: sinx}}}$

So, using these results, we get

$\rm \: = \: \dfrac{1}{2}\bigg(1 - sinx + 1 + sinx\bigg)$

$\rm \: = \: \dfrac{1}{2}\bigg(1 + 1\bigg)$

$\rm \: = \: \dfrac{1}{2}\bigg(2\bigg)$

$\rm \: = \: 1$

Hence, Proved

Additional Information :-

$\red{ \boxed{ \sf{ \:sin2x = 2sinxcosx}}}$

$\red{ \boxed{ \sf{ \:cos2x = {2cos}^{2}x - 1}}}$

$\red{ \boxed{ \sf{ \:cos2x = 1 - {2sin}^{2}x}}}$

$\red{ \boxed{ \sf{ \:cos2x = {cos}^{2}x - {sin}^{2}x}}}$

$\red{ \boxed{ \sf{ \:sin2x = \frac{2tanx}{1 + {tan}^{2} x}}}}$

$\red{ \boxed{ \sf{ \:tan2x = \frac{2tanx}{1 - {tan}^{2} x}}}}$