Question

No spam or I'll report​

Answer 1

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

Consider,

$\rm :\longmapsto\:\dfrac{1 - cos4\alpha }{ {sec}^{2} 2\alpha - 1} + \dfrac{1 + cos4\alpha }{ {cosec}^{2}2\alpha - 1}$

We know,

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

$\red{ \boxed{ \rm{ \: 1 + cos2x = {2cos}^{2}x}}}$

$\red{ \boxed{ \rm{ \: secx = \frac{1}{cosx}}}}$

$\red{ \boxed{ \rm{ \: cosecx = \frac{1}{sinx}}}}$

Using these Identities, above expression reduced to

$\rm \: = \: \: \dfrac{ {2sin}^{2}2\alpha }{\dfrac{1}{ {cos}^{2} 2\alpha } - 1} + \dfrac{ {2cos}^{2} 2\alpha }{\dfrac{1}{ {sin}^{2}2\alpha } - 1 }$

$\rm \: = \: \: \dfrac{ {2sin}^{2}2\alpha }{\dfrac{1 - {cos}^{2} 2\alpha }{ {cos}^{2} 2\alpha }} + \dfrac{ {2cos}^{2} 2\alpha }{\dfrac{1 - {sin}^{2}2\alpha }{ {sin}^{2}2\alpha }}$

We know,

$\red{ \boxed{ \rm{ \: {cos}^{2}x + {sin}^{2}x = 1}}}$

So, using this, we get

$\rm \: = \: \:\dfrac{2 {sin}^{2} 2\alpha {cos}^{2} 2\alpha }{ {sin}^{2} 2\alpha } + \dfrac{2 {sin}^{2} 2\alpha {cos}^{2} 2\alpha }{ {cos}^{2} 2\alpha }$

$\rm \: = \: \:2 {cos}^{2}2\alpha + 2 {sin}^{2}2\alpha$

$\rm \: = \: \:2 ({cos}^{2}2\alpha + {sin}^{2}2\alpha)$

$\rm \: = \: \:2 \times 1$

$\rm \: = \: \:2$

Thus,

$\red{ \boxed{\bf :\longmapsto\:\dfrac{1 - cos4\alpha }{ \: \: {sec}^{2} 2\alpha - 1 \: \: } \: + \: \dfrac{1 + cos4\alpha }{ \: \: {cosec}^{2}2\alpha - 1 \: \: } = 2}}$

Additional Information :-

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)