Question

1-cos2A is equal to: (a)sin2A (b)tan2A (c)1-sin2A (d)sec2A

Answer 1

Answer: a

Explanation: We know, by trigonometry identities,

sin2A+cos2A = 1

1-cos2A = sin2A

Answer 2

1 - cos²A is equal to sin²A. [option (a)].

• As we know, according to trigonometric identity:

                         sin²A + cos²A = 1               (1)

This can be proved as:

• Let ΔABC be a right-angled triangle with

                    ∠ABC = 90°

               so, AC is the hypotenuse

                     AB is the base

             and BC is the height

• According to the Pythagoras Theorem,

                            (AB)² + (BC)² = (AC)²          (2)

• We have

                      sinA = $\frac{BC}{AC}$ ⇒ sin²A = $\frac{BC^{2} }{AC^{2} }$

             and cosA = $\frac{AB}{AC}$ ⇒cos²A = $\frac{AB^{2} }{AC^{2} }$

• Adding both terms:

             sin²A + cos²A = $\frac{BC^{2} }{AC^{2} }$ + $\frac{AB^{2} }{AC^{2} }$

             sin²A + cos²A = $\frac{BC^{2} + AB^{2} }{AC^{2} }$

             sin²A + cos²A = $\frac{AC^{2} }{AC^{2} }$                   (from 2)

          ⇒sin²A + cos²A = 1

• Rearranging equation (1), we get:

                      1 - cos²A = sin²A

Hence, the correct option is (a) sin²A.