Question

Question 6: Prove cos¯¹ 12/13 + sin¯¹ 3/5 = sin¯¹ 56/65

Class 12 - Math - Inverse Trigonometric Functions

Answer 1

Hey there!!!

⇒ cos¯¹ 12/13 + sin¯¹ 3/5 = sin¯¹ 56/65
Taking L.H.S. :

⇒ sin¯¹ 5/13 + sin¯¹ 3/5 

Using the formula,  sin¯¹x +  sin¯¹y = sin¯¹ ( x√1-y² + y√1-x² )

⇒ sin¯¹ ( 5/13 √1-9/25 + 3 √1-25/169)
⇒ sin¯¹ ( 5/13 × 4/5 + 3/5 × 12/13)
⇒ sin¯¹ (30 + 36 / 65)
⇒ sin¯¹ (56/ 65)

Hence proved

HOPE IT HELPS...

Answer 2

Let x= cos-1(12/13)

Cosx =12/13

Sinx = root over 1-cos²x

=Root over 1-(12/13)²

=root over 1-144/169

=root over 25/169

Sinx=5/13

Again let y= sin -¹3/5

Sin y=3/5

Cos y=root over 1-sin²y

=root over 1-(3/5)²

=root over 1-9/25

=root over 16/25

Cos y = 4/5

Sin (x+y)= sinxcosy + cosxsiny

Sin(x+y)= 5/13×4/5+12/13×3/5

20/65 + 36/65

56/65

Sin(x+y) = 56/65

x+y= sin-¹ 56/65

cos-¹(12/13) + sin-¹(3/5) = sin-¹(56/65)