Trigonometry help .....?

Question

1. If tan A = n tan B , sin A = m sin B, prove that cos ^ 2 A = (m ^ 2 - 1) / (n ^ 2 - 1)
2. If cos ^ 2 a - sin ^ 2 a = tan ^ 2 b, prove that tan ^ 2 a = cos ^ 2 b - sin ^ 2 b
3. If cosec a - sin a = m and sec a - cos a = n, prove that [ { (m ^ 2) * n} ^ 2/3 ] + [ { m * ( n ^ 2 ) } ^ 2/3 ] = 1

Answer 1

Tan A = n Tan B

Divide each side by ' Tan B '

n = Tan A / Tan B

Quotient Identities is TanA = sinA/cosA

n = [sinA/cosA] / [sinB/cosB]

Then n = sinAcosB / cosAsinB

And sinA = m sinB

Divide each side by ' sin B '

m = sinA / sinB

Now take  (m²-1) / (n²-1)

= [(sinA / sinB)² -1] / [(sinAcosB / cosAsinB)²-1]

 Rewrite the expression with common denominator.

= [(sin²A - sin²B) / sin²B] / [(sin²Acos²B - cos²Asin²B) / (cos²Asin²B)]

Simplify (sin²A - sin²B)(cos²Asin²B) / (sin²B)(sin²Acos²B - cos²Asin²B)

Pythagorean Identities: sin²θ = 1 - cos²θ

= [(1 - cos²A) - (1 - cos²B)]cos²A(1 - cos²B) / (1 - cos²B)[(1 - cos²A)( cos²B) - cos²A(1 - cos²B)

= [ - cos²A  + cos²B)]cos²A(1 - cos²B) / (1 - cos²B)[(cos²B - cos²A cos²B - cos²A + cos²Acos²B

=  (cos²B - cos²A)cos²A(1 - cos²B) / (1 - cos²B) [cos²B - cos²A]

= cos²A (cos²B - cos²A)(1 - cos²B) / (1 - cos²B) [cos²B - cos²A]

= cos²A

There fore

(m² - 1) / (n² - 1) = cos²A

Answer 2

cosec a-sin a = m and sec a-cos a = n,

Recall reciprocal identities is coses A = 1/sinA and secA = 1/cosA

cosec a - sin a = m

(1/sin a) - sin a = m

Rewrite the expression with common denominator.

(1 - sin² a)/sin a = m

Pythagorean Identities is 1 - sin² θ = cos²θ

cos²a/sin a = m

m = cos²a/sin a

And sec a-cos a = n

Similarly n = sin²a/cos a

Now take [{m²×n}^2/3] + [{m×(n²)}^2/3]

m²×n  = [cos²a/sin a]²×[sin²a/cos a]

= (cos⁴a)(sin²a)/(sin²a)(cos a)

Simplify m²×n = cos³a

{m²×n}^2/3 = (cos³a)^2/3 = cos²a

m×n² = (cos²a/sin a)×(sin²a/cos a)²

= (cos²a)(sin⁴a)/(sin a)(cos²a)

Simplify m×n²=sin³a

{m×n²}^2/3 = {sin³a}^2/3 = sin²a

[{ m²×n}^2/3 + { m×n²}^2/3] = cos²a + sin²a

So, [{ m²×n}^2/3 + { m×n²}^2/3] = cos²a + sin²a = 1

There fore [{ m²×n}^2/3 + { m×n²}^2/3] = 1