**Trigonometric
Identities**

A mathematical statement of equality involving the trigonometrical ratios with one or more variable angle and is true for any value of the angle is
called a **trigonometric identity** or **trig identity**.

Generally, α, β, θ, or A, B, C symbols are used for the variable
angle in the **trigonometric identities**.

Here are some examples of **trigonometric identities**:

**Standard Trigonometric
Identities (Trig identities)**

Here are some standard **trigonometric identities** (**trig
identities**) considered as the **trigonometric formulas** as well:

**Derived Trigonometric
Relations:**

We can derive some other trigonometric relations from the
standard trigonometic identities. Here are those **derived trigonometric
relations**:

**How to Prove
Trigonometric Identities?**

To prove a trigonometric identity, we have to show that the
expressions on both the sides of equality sign are equal. For that, we take
left hand side (LHS) and simplify it to the expression equal to the right
hand side (RHS) or vice-versa.

We simplify the trigonometric expression by a process similar to
algebraic simplification. We can use almost all the algebraic formulas, methods
of factorization, rational simplification, division, multiplication, addition, and subtraction.

Moreover, we use here the **standard trigonometric identities** and
the **derived relations** as the formula while simplifying the trigonometric
expressions.

**For example:**

**Prove:** (sinθ + cosθ)^{2} = 1 + 2sinθcosθ

**Solution:** Here,

LHS = (sinθ + cosθ)^{2}

= sin^{2}θ
+ 2sinθcosθ + cos^{2}θ [∵ (a + b)^{2} = a^{2}
+ 2ab + b^{2}]

= sin^{2}θ
+ cos^{2}θ + 2sinθcosθ

= 1 + 2sinθcosθ [∵ sin^{2}θ
+ cos^{2}θ = 1]

= RHS. Proved.

**Rules for proving
trigonometric identities**

1.
Take LHS and simplify to
get RHS, if LHS is complicated.

2.
Take RHS and simplify to
get LHS, if RHS is complicated.

3.
If LHS and RHS both are
complicated, take LHS and RHS both and simplify them to a common result.

4.
If the given identity is
more complicated then change it to a simpler form by transposing or applying
the method of cross-multiplication. Then, prove the new identity by the above
procedure.

**List of Trigonometric
Formulas (Trig Formulas)**

**Proving Trigonometric
Identities**

**Worked out Examples: **

**Example 1:** Prove that: cosecA.tanA.cosA = 1

**Solution: **Here,

**Example 2:** Prove that: sec^{2}θ (1 – cos^{2}θ)
= tan^{2}θ

**Solution: **Here,

**Example 3:** Prove that: sin^{2}θ . tan^{2}θ = tan^{2}θ
– sin^{2}θ

**Solution: **Here,

**Example 4:** Prove that: tan^{4}A + tan^{2}A = sec^{4}A
– sec^{2}A

**Solution:** Here,

LHS = tan^{4}A + tan^{2}A

= tan^{2}A
(tan^{2}A + 1)

= (sec^{2}A
– 1) sec^{2}A

= sec^{4}A
– sec^{2}A

= RHS. Proved.

**Example 5:** Prove that:

**Solution: **Here,

**Example 6:**Prove that:

**Solution: **Here,

**Example 7:**Prove that:

**Solution: **Here,

**Example 8:** Prove that:

**Solution: **Here,

**Example 9:** Prove that:

**Solution: **Here,

**Example 10:** Prove that: (3 – 4sin^{2}A)(sec^{2}A –
4tan^{2}A) = (3 – tan^{2}A)(1 – 4sin^{2}A)

**Solution: **Here,

**Do you have any questions
or problems regarding the Trigonometric Identities?**

You can ask here, in the comment section below.

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