Trigonometry is basically measuring the length and the angles of triangles. Trigonometry has a lot of formulas using the trigonometric functions, i.e., sine, cosine, tangent, secant, cosecant and cotangent. These trigonometry functions involved in an expression are used by the trigonometry identities to solve problems.

Below are some of the trigonometry formulas given:

1. $cos^{2}(x)$ + $sin^{2}(x) $  = 1

2. 1 + $tan^{2}(x)$ = $sec^{2}(x)$

3. 1 + $cot^{2}(x)$ = $cosec^{2}(x)$

4. $sin(x \pm y)$ = $sin(x)cos(y)\pm cos(x)sin(y)$

5. $cos(x \pm y)$ = $cos(x)cos(y)\pm sin(x)sin(y)$

6. $sin(x)sin(x)$ = $\frac{1}{2}$$[cos(x-y)-cos(x+y)]$

7. $cos(x)cos(x)$ = $\frac{1}{2}$$[cos(x-y)+cos(x+y)]$

8. $cos(x)+cos(x)$ = 2$cos$ $\frac{x+y}{2}$$cos$ $\frac{x-y}{2}$

9. $sin(x)-sin(x)$ = 2$sin$ $\frac{x-y}{2}$$cos$ $\frac{x+y}{2}$

Below are some of the solved problems based on trigonometry:

Question 1: Simplify: $sin^{2}x$ + $sin^{2}x \ cot^{2}x$
Solution:
Simplifying,
$sin^{2}x$ + $sin^{2}x \ cot^{2}x$ = $sin^{2}x$ (1 + $cot^{2}x$)

We know that, 1 + $cot^{2}(x)$ = $cosec^{2}x$ and $cosec \ x$ = $\frac{1}{sin \ x}$

Simplifying further,
$sin^{2}x$ (1 + $cot^{2}x$) = $sin^{2}x$ . $cosec^{2}x$
= $\not{sin^{2}x}$ . $\frac{1}{\not{sin^{2}x}}$ = 1

Therefore, 
$sin^{2}x$ + $sin^{2}x \ cot^{2}x$ = 1.

Question 2: Simplify: $sin^{4}x$ - $cos^{4}x$
Solution:
Simplifying,
$sin^{4}x$ - $cos^{4}x$
= ($sin^{2}x$ + $cos^{2}x$)($sin^{2}x$ - $cos^{2}x$)

We know that, As, $cos^{2}(x)$ + $sin^{2}(x) $  = 1

Simplifying further,
($sin^{2}x$ + $cos^{2}x$)($sin^{2}x$ - $cos^{2}x$)
= 1 . ($sin^{2}x$ - $cos^{2}x$)   
= $sin^{2}x$ - (1 - $sin^{2}x$)   
= $sin^{2}x$ - 1 + $sin^{2}x$)   
= 2$sin^{2}x$ - 1

Therefore,$sin^{4}x$ - $cos^{4}x$ = 2$sin^{2}x$ - 1.