Question

Prove the above please​

Answer 1

$\huge\mathfrak\green{☟ \: \: \: answer \: \: \: \: ✎}$

Here we need to show that L.H.S = 1

Before that we need to use two important identity

i.e

$\cos(a \times b) \times \cos(a - b) = {cos}^{2} a - {sin}^{2} b$

‎‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎and

$\sin(a + b) \times \sin(a - b) = {sin}^{2} a - {sin}^{2} b$

$\sf \colorbox{pink} {\red(Now, }$

$L.H.S = \cot( \frac{\pi}{4} + x) \cot( \frac{\pi}{4} - x ) = \frac{ \cos( \frac{\pi}{4} + x ) }{ \sin( \frac{\pi}{4} + x) } \times \frac{ \cos( \frac{\pi}{4} - x) }{ \sin( \frac{\pi}{4} - x) }$

$= \frac{ {cos}^{2} \frac{\pi}{4} - {sin}^{2} x}{ {sin}^{2} \frac{\pi}{4} - {sin}^{2}x } = \frac{( \frac{1}{ \sqrt{2} ^{} } {)}^{2} - {sin}^{2} x}{( \frac{1}{ \sqrt{2} } {)}^{2} - {sin}^{2} x} = \frac{ \frac{1}{2} - {sin}^{2} x}{ \frac{1}{2} - {sin}^{2} x }$

$= 1 = R.H.S$

Hence Proved.

$\\ \\ \\ \sf \colorbox{gold} {\red(ANSWER ᵇʸ ⁿᵃʷᵃᵇ⁰⁰⁰⁸} \\ and\\ \sf \colorbox{lightgreen} {\red❤ANSWER ᵇʸ ᶠˡⁱʳᵗʸ ᵇᵒʸ}$

Answer 2

Answer:

Here we need to show that L.H.S = 1

Before that we need to use two important identity

i.e

\cos(a \times b) \times \cos(a - b) = {cos}^{2} a - {sin}^{2} bcos(a×b)×cos(a−b)=cos

2

a−sin

2

b

‎‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎‎ ‎and

\sin(a + b) \times \sin(a - b) = {sin}^{2} a - {sin}^{2} bsin(a+b)×sin(a−b)=sin

2

a−sin

2

b

\sf \colorbox{pink} {\red(Now, }

(Now,

L.H.S = \cot( \frac{\pi}{4} + x) \cot( \frac{\pi}{4} - x ) = \frac{ \cos( \frac{\pi}{4} + x ) }{ \sin( \frac{\pi}{4} + x) } \times \frac{ \cos( \frac{\pi}{4} - x) }{ \sin( \frac{\pi}{4} - x) }L.H.S=cot(

4

π

+x)cot(

4

π

−x)=

sin(

4

π

+x)

cos(

4

π

+x)

×

sin(

4

π

−x)

cos(

4

π

−x)

= \frac{ {cos}^{2} \frac{\pi}{4} - {sin}^{2} x}{ {sin}^{2} \frac{\pi}{4} - {sin}^{2}x } = \frac{( \frac{1}{ \sqrt{2} ^{} } {)}^{2} - {sin}^{2} x}{( \frac{1}{ \sqrt{2} } {)}^{2} - {sin}^{2} x} = \frac{ \frac{1}{2} - {sin}^{2} x}{ \frac{1}{2} - {sin}^{2} x }=

sin

2

4

π

−sin

2

x

cos

2

4

π

−sin

2

x

=

(

2

1

)

2

−sin

2

x

(

2

1

)

2

−sin

2

x

=

2

1

−sin

2

x

2

1

−sin

2

x

= 1 = R.H.S=1=R.H.S

Hence Proved.

Step-by-step explanation:

hope it helps you!

thank you!