Prove the trigonometric identity: $1+\cot ^2 A=\operatorname{cosec}^2 A$
Answer (2)
Start with the definitions: cot A = s i n A c o s A and csc A = s i n A 1 .
Substitute into the left-hand side: 1 + cot 2 A = 1 + s i n 2 A c o s 2 A .
Find a common denominator: 1 + s i n 2 A c o s 2 A = s i n 2 A s i n 2 A + c o s 2 A .
Apply the Pythagorean identity: s i n 2 A s i n 2 A + c o s 2 A = s i n 2 A 1 = csc 2 A . Thus, 1 + cot 2 A = csc 2 A .
The identity is proven: 1 + cot 2 A = csc 2 A .
Explanation
Understanding the Problem We are tasked with proving the trigonometric identity 1 + cot 2 A = csc 2 A . To do this, we will start with the definitions of cot A and csc A in terms of sin A and cos A , and then manipulate the left-hand side of the equation until it matches the right-hand side.
Substituting Definitions We know that cot A = s i n A c o s A and csc A = s i n A 1 . Let's substitute these definitions into the left-hand side of the identity: 1 + cot 2 A = 1 + ( sin A cos A ) 2 = 1 + sin 2 A cos 2 A .
Finding a Common Denominator To combine the terms on the left-hand side, we need a common denominator. We can rewrite 1 as s i n 2 A s i n 2 A : 1 + sin 2 A cos 2 A = sin 2 A sin 2 A + sin 2 A cos 2 A .
Combining Fractions Now, we can combine the fractions: sin 2 A sin 2 A + sin 2 A cos 2 A = sin 2 A sin 2 A + cos 2 A .
Applying the Pythagorean Identity We can now use the Pythagorean identity, which states that sin 2 A + cos 2 A = 1 : sin 2 A sin 2 A + cos 2 A = sin 2 A 1 .
Rewriting the Expression We can rewrite s i n 2 A 1 as ( s i n A 1 ) 2 : sin 2 A 1 = ( sin A 1 ) 2 .
Substituting the Definition of Cosecant Finally, we substitute the definition of csc A = s i n A 1 back into the equation: ( sin A 1 ) 2 = csc 2 A .
Conclusion Thus, we have shown that 1 + cot 2 A = csc 2 A .
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and navigation. For example, in physics, they are used to simplify equations involving oscillatory motion, such as the motion of a pendulum or the propagation of waves. In engineering, they are used in the design of structures and circuits. In navigation, they are used to calculate distances and angles.
The identity 1 + cot 2 A = csc 2 A is proven by substituting definitions and using the Pythagorean identity to manipulate the left-hand side until it matches the right-hand side. This process involves rewriting terms with a common denominator and applying trigonometric identities. Therefore, we conclude that the identity is valid.
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