Trigonometric identities are equations that show relationships between trigonometric functions that are used to simplify trigonometric equations.
Verifying a trigonometric identity involves proving that both sides of an equation are always equal, no matter the values of the variables. For example, the identity: sin2 θ + cos2 θ = 1, is true for all values of θ
To prove an identity, we mostly start with one side of the equation and then simplify it using the known trigonometric identities till we get to the other side of the equation. Sometimes, we need to simplify both sides of the equation to get to a common term(s) that cannot be simplified even further.
Here is the list of the fundamental trigonometric identities and the types we use to verify them.
Now, let us verify a few such identities.
Verify: 1 + tan2 θ = sec2 θ
Step 1: Using Pythagorean Identity
First, we start by identifying the identity which is needed. Here, we will use the Pythagorean identity:
sin2 θ + cos2 θ = 1
Step 2: Dividing Both Sides
Now, we will divide both sides of the equation to make it the same as the left side of the given equation.
On dividing by cos2 θ,
⇒ ${\dfrac{\sin ^{2}\theta }{\cos ^{2}\theta }+\dfrac{\cos ^{2}\theta }{\cos ^{2}\theta }=\dfrac{1}{\cos ^{2}\theta }}$
Step 3: Using Reciprocal Identity
Again, we will identify the identity that is needed to simplify the equation further. Here, we will apply reciprocal identity.
⇒ ${\tan ^{2}\theta +1=\sec ^{2}\theta}$ (Since tan θ = ${\dfrac{\sin \theta }{\cos \theta }}$ and ${\sec \theta =\dfrac{1}{\cos \theta }}$)
Thus, 1 + tan2 θ = sec2 θ
Verify: ${\dfrac{\sin \theta }{1+\cos \theta }=\dfrac{1-\cos \theta }{\sin \theta }}$
Step 1: Considering the Left-Hand Side of the Equation
Since the left part is a bit more complicated, we will start by simplifying that part to get to the other side of the equation.
${\dfrac{\sin \theta }{1+\cos \theta }}$
Step 2: Rationalizing the Denominator
Since the denominator is a little bit complicated, we will multiply the numerator and denominator by the conjugate of the denominator (rationalization).
On multiplying the numerator and the denominator by the conjugate of 1 + cos θ,
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) }{\left( 1+\cos \theta \right) \left( 1-\cos \theta \right) }}$
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) }{1-\cos ^{2}\theta }}$
Step 3: Using Pythagorean Identity
Now, we will use the identity which is needed to simplify expression further. Here, we will use the Pythagorean identity.
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) }{\sin ^{2}\theta }}$ (Since the Pythagorean identity: sin2 θ + cos2 θ = 1)
Step 4: Canceling Out
Here, canceling out sin θ from the numerator and denominator, we get
= ${\dfrac{1-\cos \theta }{\sin \theta }}$
This matches the right-hand side, and thus, the identity is verified.
Verify: ${\dfrac{1}{\sin \theta }-\sin \theta =\cot \theta \cos \theta}$
Step 1: Considering the Left-Hand Side of the Equation
Since the left part is a bit more complicated, we will start by simplifying that part to get to the other side of the equation.
${\dfrac{1}{\sin \theta }-\sin \theta}$
Step 2: Rewriting the Expression With the Common Denominator
To simplify the expression, first, we will find the common denominator and then rewrite the expression.
= ${\dfrac{1}{\sin \theta }-\dfrac{\sin \theta }{1}}$
Step 3: Combining the Common Terms and Adding the Fractions
Now, we will combine the common terms to add the fractions.
= ${\dfrac{1-\sin ^{2}\theta }{\sin \theta }}$
Step 4: Using Pythagorean Identity
Next, we will identify which identity is needed to simplify expression further. Here, we will use the Pythagorean identity.
= ${\dfrac{\cos ^{2}\theta }{\sin \theta }}$ (Since the Pythagorean identity: sin2 θ + cos2 θ = 1)
Step 5: Factoring the Fraction
Now, we will factor out the common terms from the numerator and denominator.
= ${\dfrac{\cos \theta }{\sin \theta }\cdot \cos \theta}$
Step 6: Using Reciprocal Identity
Again, we will use the appropriate identity to simplify the equation further. Here, we will apply reciprocal identity.
= ${\cot \theta \cos \theta}$ (Since the reciprocal identity: cot θ = ${\dfrac{\cos \theta }{\sin \theta }}$)
Thus, ${\dfrac{1}{\sin \theta }-\sin \theta =\cot \theta \cos \theta}$
Prove: ${\dfrac{\sec \theta }{\text{cosec}\, \theta }=\dfrac{\sin \theta }{\cos \theta }}$
Step 1: Considering the Left-Hand Side
Since the left part is a bit more complicated, we will start by simplifying that part to get to the other side of the equation.
${\dfrac{\sec \theta }{\text{cosec}\, \theta }}$
= ${\sec \theta \cdot \dfrac{1}{\text{cosec}\, \theta }}$ …..(i)
Step 2: Using the Definitions of Reciprocal Functions
Here, the right side of the equation contains sin θ and cos θ. Thus, we will rewrite the terms of the left side in terms of sin θ and cos θ to get the other side of the equation.
As we know from the definitions of sec θ and cosec θ,
${\sec \theta =\dfrac{1}{\cos \theta }}$ and ${\text{cosec}\, \theta =\dfrac{1}{\sin \theta }}$
Now, from (i), we get
= ${\dfrac{1}{\cos \theta }\cdot \sin \theta}$
= ${\dfrac{\sin \theta }{\cos \theta }}$
Thus, ${\dfrac{\sec \theta }{\text{cosec}\, \theta }=\dfrac{\sin \theta }{\cos \theta }}$
Solved Examples
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Prove: ${\dfrac{\sec ^{2}\theta -1}{\sec ^{2}\theta }=\sin ^{2}\theta}$
Solution:
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Step 1: Considering the Left-Hand Side
${\dfrac{\sec ^{2}\theta -1}{\sec ^{2}\theta }}$
Step 2: Using Pythagorean Identity
As we know, the Pythagorean identity is: 1 + tan2 θ = sec2 θ ⇒ tan2 θ = sec2 θ – 1
= ${\dfrac{\tan ^{2}\theta }{\sec ^{2}\theta }}$
Step 3: Using Quotient Identity
As we know, tan θ = ${\dfrac{\sin \theta }{\cos \theta }}$ and ${\sec \theta =\dfrac{1}{\cos \theta }}$
= ${\dfrac{\dfrac{\sin ^{2}\theta }{\cos ^{2}\theta }}{\dfrac{1}{\cos ^{2}\theta }}}$
Step 4: Canceling Out
= ${\sin ^{2}\theta}$
Thus, the left-hand side equals the right-hand side:
${\dfrac{\sec ^{2}\theta -1}{\sec ^{2}\theta }=\sin ^{2}\theta}$
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Verify: ${\dfrac{\cos ^{2}\theta -\sin ^{2}\theta }{\cos ^{2}\theta +\sin ^{2}\theta }=\cos 2\theta}$
Solution:
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Step 1: Considering the Left-Hand Side
${\dfrac{\cos ^{2}\theta -\sin ^{2}\theta }{\cos ^{2}\theta +\sin ^{2}\theta }}$
Step 2: Using Pythagorean Identity
As we know, the Pythagorean identity is: sin2 θ + cos2 θ = 1
= ${\dfrac{\cos ^{2}\theta -\sin ^{2}\theta }{1}}$
Step 3: Using Double-Angle Identity
As we know, the double-angle identity for cosine is: cos 2θ = cos2 θ – sin2 θ
= ${\cos 2\theta}$
Thus, the left-hand side equals the right-hand side:
${\dfrac{\cos ^{2}\theta -\sin ^{2}\theta }{\cos ^{2}\theta +\sin ^{2}\theta }=\cos 2\theta}$
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Verify: ${\dfrac{\sin \theta }{1+\cos \theta }+\dfrac{\sin \theta }{1-\cos \theta }=2\text{cosec}\, \theta}$
Solution:
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Step 1: Considering the Left-Hand Side
${\dfrac{\sin \theta }{1+\cos \theta }+\dfrac{\sin \theta }{1-\cos \theta }}$
Step 2: Making the Denominators the Same
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) }{\left( 1+\cos \theta \right) \left( 1-\cos \theta \right) }+\dfrac{\sin \theta \left( 1+\cos \theta \right) }{\left( 1-\cos \theta \right) \left( 1+\cos \theta \right) }}$
Step 3: Adding the Fractions
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) +\sin \theta \left( 1+\cos \theta \right) }{\left( 1+\cos \theta \right) \left( 1-\cos \theta \right) }}$
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) +\sin \theta \left( 1+\cos \theta \right) }{1-\cos ^{2}\theta }}$
Step 4: Using Pythagorean Identity
As we know, the Pythagorean identity is: sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ
= ${\dfrac{\sin \theta \left( 1-\cos \theta \right) +\sin \theta \left( 1+\cos \theta \right) }{\sin ^{2}\theta }}$
= ${\dfrac{\sin \theta -\sin \theta \cos \theta +\sin \theta +\sin \theta \cos \theta }{\sin ^{2}\theta }}$
Step 5: Canceling Out
= ${\dfrac{2\sin \theta }{\sin ^{2}\theta }}$
= ${\dfrac{2}{\sin \theta }}$
As we know, the reciprocal identity is: ${\text{cosec}\, \theta =\dfrac{1}{\sin \theta }}$
= ${2\text{cosec}\, \theta}$
Thus, the left-hand side equals the right-hand side:
${\dfrac{\sin \theta }{1+\cos \theta }+\dfrac{\sin \theta }{1-\cos \theta }=2\text{cosec}\, \theta}$
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Prove: cot2 θ + cosec2 θ = 2cosec2 θ – 1
Solution:
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Step 1: Considering the Left-Hand Side
cot2 θ + cosec2 θ
Step 2: Using Pythagorean Identity
As we know, the Pythagorean identity is: 1 + cot2 θ = cosec2 θ ⇒ cot2 θ = cosec2 θ – 1
= cosec2 θ – 1 + cosec2 θ
= 2cosec2 θ – 1
Thus, the left-hand side equals the right-hand side:cot2 θ + cosec2 θ = 2cosec2 θ – 1
