Conditional Trigonometric Identities

Conditional Identities

Conditional trigonometric identities are special equations in trigonometry that hold true only under specific conditions. Unlike standard trigonometric identities such as $\sin^2 \theta + \cos^2 \theta = 1$, these identities are valid when certain values or relations between angles are satisfied.

For example, a conditional identity may require $\sin \theta \neq 0$ or $\cos \theta \neq 0$ for the formula to be true. Such conditional identities in trigonometry help students solve higher-level problems, prove equations, and connect algebraic expressions with trigonometric ratios. Understanding their meaning and scope is essential for mastering the NCERT syllabus topics, board exams, and advanced entrance test preparation.

List of Conditional Trigonometric Identities

Below is the list of conditional trigonometric identities that are derived using the condition $A + B + C = \pi$. These identities are important for Class 11, Class 12, and JEE exams, making problem-solving in trigonometry easier and faster.

Using the Condition $A + B + C = \pi$

In any triangle $ABC$, the sum of angles is $\pi$ (or $180^\circ$). This condition forms the basis of several conditional trigonometric identities useful in Class 11, Class 12, and JEE problems.

Here, the condition is that $A, B,$ and $C$ are the angles of $\triangle ABC$, so:

$A + B + C = \pi$

From this, we can write:

• $A + B = \pi - C$

• $A + C = \pi - B$

• $B + C = \pi - A$

Using these relations, we derive important conditional identities in trigonometry:

Conditional Identities Formula for Sine and Cosine

In conditional trigonometric identities, sine and cosine play the most fundamental role. When the condition $A + B + C = \pi$ holds true in a triangle, we derive the following identities:

• $\sin(A + B) = \sin C$

• $\sin(A + C) = \sin B$

• $\sin(B + C) = \sin A$

Similarly, for cosine:

• $\cos(A + B) = -\cos C$

• $\cos(A + C) = -\cos B$

• $\cos(B + C) = -\cos A$

These conditional sine and cosine identities are widely used in Class 11 and Class 12 trigonometry, as well as in JEE Main and Advanced problems.

Conditional Identities Formula for Tan and Cot

Tangent and cotangent also have special conditional identities derived from the same condition $A + B + C = \pi$:

• $\tan(A + B) = -\tan C$

• $\tan(A + C) = -\tan B$

• $\tan(B + C) = -\tan A$

For cotangent:

• $\cot(A + B) = -\cot C$

• $\cot(A + C) = -\cot B$

• $\cot(B + C) = -\cot A$

These are especially important in competitive exam trigonometry questions, where simplification using conditional tan and cot identities saves time.

Conditional Identities Formula for Sec and Cosec

Secant and cosecant identities also exist under the same condition, and they are useful for proving equations or solving advanced problems:

• $\sec(A + B) = -\sec C$

• $\sec(A + C) = -\sec B$

• $\sec(B + C) = -\sec A$

Similarly, for cosecant:

• $\csc(A + B) = \csc C$

• $\csc(A + C) = \csc B$

• $\csc(B + C) = \csc A$

These conditional sec and cosec identities extend the standard trigonometric formulas and are frequently tested in board exams and entrance exams.

Examples of Conditional Trigonometric Identities

Below are examples of conditional trigonometric identities that hold true only under specific conditions such as $A + B + C = \pi$.

Example 1 – Relation Between $\tan A, \tan B,$ and $\tan C$

If $A + B + C = \pi$, then:

$\tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C$

Proof:

$A + B = \pi - C$
$\Rightarrow \tan(A + B) = \tan(\pi - C)$
$\Rightarrow \frac{\tan A + \tan B}{1 - \tan A \tan B} = -\tan C$
$\Rightarrow \tan A + \tan B = -\tan C + \tan A \tan B \tan C$
$\Rightarrow \tan A + \tan B + \tan C = \tan A \tan B \tan C$

Example 2 – Half-Angle Identity

$\tan \frac{A}{2} \tan \frac{B}{2} + \tan \frac{B}{2} \tan \frac{C}{2} + \tan \frac{C}{2} \tan \frac{A}{2} = 1$

Proof:

$A + B + C = \pi \Rightarrow \frac{A}{2} + \frac{B}{2} = \frac{\pi}{2} - \frac{C}{2}$
$\Rightarrow \tan\left(\frac{A}{2} + \frac{B}{2}\right) = \tan\left(\frac{\pi}{2} - \frac{C}{2}\right) = \cot \frac{C}{2}$
$\Rightarrow \frac{\tan \frac{A}{2} + \tan \frac{B}{2}}{1 - \tan \frac{A}{2} \tan \frac{B}{2}} = \frac{1}{\tan \frac{C}{2}}$
$\Rightarrow \tan \frac{A}{2} \tan \frac{C}{2} + \tan \frac{B}{2} \tan \frac{C}{2}$ $= 1 - \tan \frac{A}{2} \tan \frac{B}{2}$
$\Rightarrow \tan \frac{A}{2} \tan \frac{B}{2} + \tan \frac{B}{2} \tan \frac{C}{2} + \tan \frac{C}{2} \tan \frac{A}{2} = 1$

Example 3 – Identity Involving $\sin A + \sin B + \sin C$

$\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$

Proof:

$\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$

Since $\frac{A + B}{2} = \frac{\pi - C}{2} = \frac{\pi}{2} - \frac{C}{2}$, we get:

$\sin \left(\frac{A + B}{2}\right) = \cos \frac{C}{2}$

Also, $\sin C = 2 \sin \frac{C}{2} \cos \frac{C}{2}$

Thus,

$\sin A + \sin B + \sin C = 2 \cos \frac{C}{2}\left(\cos \frac{A - B}{2} + \sin \frac{C}{2}\right)$
$= 2 \cos \frac{C}{2}\left(\cos \frac{A - B}{2} + \cos \frac{A + B}{2}\right)$
$= 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$

Example 4 – Identity Involving $\sin^2 A + \sin^2 B + \sin^2 C$

If $A + B + C = \pi$, then:

$\sin^2 A + \sin^2 B + \sin^2 C = 4 \sin A \sin B \sin C$

Proof:

Using $\cos(A + B) = -\cos C$, we write:

$\sin^2 A + \sin^2 B + \sin^2 C$
$= 2 \sin(A + B) \cos(A - B) + \sin^2 C$
$= 2 \sin C \cos(A - B) + 2 \sin C \cos C$
$= 2 \sin C \left(\cos(A - B) - \cos(A + B)\right)$
$= 4 \sin A \sin B \sin C$

Thus, the conditional identity:

$\sin^2 A + \sin^2 B + \sin^2 C = 4 \sin A \sin B \sin C$

is valid whenever $A + B + C = \pi$.

Solved Examples Based on Conditional Identities

Example 1: If $A, B$, and $C$ is the angles of triangles then Simplify the function

$
\sin (A+B) \cos (A+B)+\sin (B+C) \cos (B+C)+\sin (C+A)
$

1) $2 \sin A \cdot \sin B \cdot \sin C$
2) $\sin A \cdot \sin B \cdot \sin C$
3) $4 \sin A \cdot \sin B \cdot \sin C$
4) $2 \sin A+2 \sin B+2 \sin C$

Solution:

Given $\mathrm{A}, \mathrm{B}, \mathrm{C}$ is the angles of triangles so $\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$

$
\begin{aligned}
& \sin (A+B) \cos (A+B)+\sin (B+C) \cos (B+C)+\sin (C+A \\
& -\sin (C) \cos (C)-\sin (A) \cos (A)-\sin (B) \cos (B) \\
& \{\text { if } \mathrm{A}+\mathrm{B}+\mathrm{C}=\pi \text { then } \\
& \sin (A+B)=\sin (C) \text {, and } \cos (A+B)=-\cos (C)\} \\
& \sin (A+B) \cos (A+B)+\sin (B+C) \cos (B+C)+\sin (C+A \\
& \frac{1}{2}\{2 \sin (C) \cos (C)+2 \sin (A) \cos (A)+2 \sin (B) \cos (B)\} \\
& =\frac{1}{2}\{\sin 2 C+\sin 2 A+\sin 2 B\} \\
& =\frac{1}{2}\{4 \sin A \sin B \sin C\}\{\text { by above concept }\} \\
& =2 \sin A \sin B \sin C
\end{aligned}
$
Hence, the answer is option 4.

Example 2: If $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ is the angles of a triangle, and $\tan A+\tan B+\tan C=3 \sqrt{3}$, then which of the following is true?
1) Three different triplets of angles are possible
2) $A B C$ is a right triangle
3) $A B C$ can be an equilateral triangle but not necessarily
4) $A B C$ is an equilateral triangle

Solution:
Given $\tan A+\tan B+\tan C=3 \sqrt{3}$
we know

$
\begin{aligned}
& \tan A+\tan B+\tan C=\tan A \tan B \tan C \quad\{\text { for triangle } \mathrm{ABC}\} \\
& A \cdot M \cdot \geq G \cdot M \\
& \frac{\tan A+\tan B+\tan C}{3} \geq(\tan A \tan B \tan C)^{\frac{1}{3}} \\
& \sqrt{3} \geq(3 \sqrt{3})^{\frac{1}{3}} \\
& A \cdot M .=G \cdot M . \text { only possible when } \tan A=\tan B=\tan C \text { or } \mathrm{A}=\mathrm{B}=\mathrm{C} \\
& A+B+C=\pi \\
& \text { so } A=B=C=\frac{\pi}{3}
\end{aligned}
$

Hence, the answer is option 4.

Example 3: Number of solutions of $\tan x+\tan 3 x+\tan 5 x=\tan x \tan 3 x \tan 5 x$ for $x \in\left[0, \frac{\pi}{2}\right]$ is?
1) 3
2) 4
3) 5
4) 6

Solution:

Conditional Identities -
Here the condition is that $A, B$, and $C$ are the angles of triangle $A B C$, and $A+B+C=\pi$.
As, $A+B+C=\pi$ then, $A+B=\pi-C, A+C=\pi-B$ and $B+C=\pi-A$
Using the above conditions, we can get some important identities.

$
\text { 1. } \sin (A+B)=\sin (\pi-C)=\sin C
$
Similarly, $\sin (A+C)=\sin B$ and $\sin (B+C)=\sin A$

$
\text { 2. } \cos (A+B)=\cos (\pi-C)=-\cos C
$
Similarly, $\cos (A+C)=-\cos B$ and $\cos (B+C)=-\cos A$

$
\text { 3. } \tan (A+B)=\tan (\pi-C)=-\tan C
$
Similarly, $\tan (A+C)=-\tan B$ and $\tan (B+C)=-\tan A$

$
\begin{aligned}
& \quad \tan x+\tan 3 x+\tan 5 x=\tan x \tan 3 x \tan 5 x \text { is possible when } \mathrm{x}+3 \mathrm{x}+5 \mathrm{x}=n \pi \\
& x=\frac{n \pi}{9} \\
& x=\left\{\frac{\pi}{9}, \frac{2 \pi}{9}, \frac{\pi}{3}, \frac{4 \pi}{9}\right\} \\
& \text { Number of solution is } 4
\end{aligned}
$
Example 4: In $\triangle A B C$, $\tan A+\tan B+\tan C$ is equal to.

Solution: Given that,

$\tan A+\tan B+\tan C$

$A+B+C=\pi$

$A+B=\pi-C$

Multiplying by tan on both sides to get,

$\tan (A+B)=\tan (\pi-C)$

$\frac{\tan A+\tan B}{1-\tan A \tan B}=-\tan C$

$\tan A+\tan B=-\tan C(1-\tan A \tan B)$

$\tan A+\tan B=-\tan C+\tan C \tan A \tan B$

$\tan A+\tan B+\tan C=\tan A \tan B \tan C$

Example 5: If $\sin ^{-1} a+\sin ^{-1} b+\sin ^{-1} c=\pi$, then the value of $a \sqrt{\left(1-a^2\right)}+b \sqrt{\left(1-b^2\right)}+c \sqrt{\left(1-c^2\right)}$ will be
1) $2 a b c$
2) $a b c$
3) $\frac{1}{2} a b c$
4) $\frac{1}{3} a b c$

Solution:

Double Angle Formula -

$\begin{aligned} \sin 2 \alpha & =2 \sin \alpha \cos \alpha \\ \cos 2 \alpha & =\cos ^2 \alpha-\sin ^2 \alpha \\ & =2 \cos ^2 \alpha-1 \\ & =1-2 \sin ^2 \alpha \\ \tan 2 \alpha & =\frac{2 \tan \alpha}{1-\tan ^2 \alpha}\end{aligned}$

- wherein

These are formulae for double angles
$
\begin{aligned}
& \text { Let } \sin ^{-1} a=A \\
& \quad \sin ^{-1} b=B \\
& \quad \sin ^{-1} c=C \\
& \therefore \sin A=a, \sin B=b, \sin C=c
\end{aligned}
$

and $A+B+C=\pi$, then

$
\sin 2 A+\sin 2 B+\sin 2 C=4 \sin A \sin B \sin C
$
Now $a \sqrt{\left(1-a^2\right)}+b \sqrt{\left(1-c^2\right)}+c \sqrt{\left(1-c^2\right)}$

$
\begin{aligned}
& =\sin A \cos A+\sin B \cos B+\sin C \cos C \\
& =\frac{1}{2}[\sin 2 A+\sin 2 B+\sin 2 C]=2 \sin A \sin B \sin C=2 a b c
\end{aligned}
$

List of Topics Related to the Conditional Trigonometric Identities

Below is the list of topics related to Conditional Trigonometric Identities that will help you cover important formulas, examples, and applications for Class 11, Class 12, and competitive exams like JEE.

NCERT Resources

Below are NCERT resources for Class 11 Chapter 3 – Trigonometric Functions, including notes, NCERT solutions, and Exemplar solutions to help students understand concepts, practice problems, and prepare for exams effectively.

NCERT Class 11 Chapter 3 - Trigonometric Functions Notes

NCERT Class 11 solutions for Chapter 3 - Trigonometric Functions

NCERT Exemplar solutions for Class 11 Chapter 3 - Trigonometric Functions

Practice Questions on Conditional Trigonometric Identities

Below are some practice questions on Conditional Trigonometric Identities that will help you strengthen your concepts, improve problem-solving skills, and prepare effectively for Class 11, Class 12, and JEE exams.

Conditional Trigonometric Identities - Practice Question

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