Question

CosA=12/13find 1/1+sin^25A+1/1+cos^25A

Answer 1

Given:

cosA = 12/13

To Find:

1/(1+sin²5A) + 1/(1+cos²5A)

Solution:

First lets find the value of sin5A .

We know,

• sin5A = sin (2A + 3A)
• sin (2A + 3A) = sin2Acos3A + cos2Asin3A

Given cos A = 12/13

• sin A = 5/13

We know sin2A = 2sinAcosA

• sin2A = 2 x 5 x 12 / 13 x 13 = sin2A = 120/169
• cos2A = √1-sin²2A = √1 - 120²/169² =√ (169-120)(120+169)/169
• cos2A = √49x289/169 = 7x17/169 = 119/169

We know sin3A = sin(2A+A) = sin2AcosA + cos2AsinA

• sin3A = 120 x 12/169 x 13 + 119 x 5/169 x 13
• sin3A  =(1440 + 595 )/13³
• sin3A = 2035/2197
• cos3A = √1-sin²3A = √(2197-2035)(2197+2035)/2197
• cos3A = 828/13³

Now Lets find sin5A = sin3Acos2A + sin2Acos3A = 2035x119/$13^{5}$ + 120x828/$13^{5}$

• sin5A = (242165 + 99360)/371293  =341525/371293 = 0.9198

• cos5A = √1- sin²5A = √(371293-341525)(371293+341525)/371293

• cos5A = 145668/371293 = 0.39

Therefore sin10A = 2sin5Acos5A

• sin5Acos5A = 0.36

Now

• 1/(1+sin²5A) + 1/(1+cos²5A) = (1+ cos²5A + 1 + sin²5A)/(1+sin²5A)(1+cos²5A) =>
• 3/(1 + sin²5A + cos²5A + (sin5Acos5A)²
• ==>
• 3/(2+0.1296)
• 3/2.1296
• The answer = 1.41

Therefore 1/(1+sin²5A) + 1/(1+cos²5A) = 1.41