## anonymous one year ago cot x sec4x = cot x + 2 tan x + tan3x

1. anonymous

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.

2. Michele_Laino

I think that we have to use these identities: $\Large \begin{gathered} {\left( {\sec x} \right)^4} = \frac{1}{{{{\left( {\cos x} \right)}^4}}} \hfill \\ \hfill \\ {\left( {\tan x} \right)^3} = {\left( {\frac{{\sin x}}{{\cos x}}} \right)^3} \hfill \\ \end{gathered}$

3. anonymous

ok

4. Michele_Laino

so, left side is: $\Large \cot x{\left( {\sec x} \right)^4} = \frac{{\cos x}}{{\sin x}}\frac{1}{{{{\left( {\cos x} \right)}^4}}} = ...?$ please simplify

5. anonymous

$\cos x/\sin x \times \left( cosx \right)^4$

6. Michele_Laino

please you have to simplify cos(x) with (cos x)^4

7. anonymous

1/sin x * (cosx)^3

8. Michele_Laino

ok! correct!

9. Michele_Laino

now, we have to make the right side, equal to your expression for left side

10. Michele_Laino

so, we can write: $\large \cot x + 2\tan x + {\left( {\tan x} \right)^3} = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + {\left( {\frac{{\sin x}}{{\cos x}}} \right)^3}$

11. Michele_Laino

or: $\large \begin{gathered} \cot x + 2\tan x + {\left( {\tan x} \right)^3} = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + {\left( {\frac{{\sin x}}{{\cos x}}} \right)^3} = \hfill \\ \hfill \\ = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + \frac{{{{\left( {\sin x} \right)}^3}}}{{{{\left( {\cos x} \right)}^3}}} \hfill \\ \end{gathered}$

12. Michele_Laino

what is the common denominator?

13. anonymous

sinx*cosx^3

14. Michele_Laino

correct!

15. Michele_Laino

so, we can write this: $\large \begin{gathered} \cot x + 2\tan x + {\left( {\tan x} \right)^3} = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + {\left( {\frac{{\sin x}}{{\cos x}}} \right)^3} = \hfill \\ \hfill \\ = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + \frac{{{{\left( {\sin x} \right)}^3}}}{{{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{{\cos x{{\left( {\cos x} \right)}^3} + 2\sin x\sin x{{\left( {\cos x} \right)}^2} + \sin x{{\left( {\sin x} \right)}^3}}}{{\sin x{{\left( {\cos x} \right)}^3}}} \hfill \\ \end{gathered}$

16. anonymous

ok I get it

17. Michele_Laino

we can simplify like this: $\large \begin{gathered} = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + \frac{{{{\left( {\sin x} \right)}^3}}}{{{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{{\cos x{{\left( {\cos x} \right)}^3} + 2\sin x\sin x{{\left( {\cos x} \right)}^2} + \sin x{{\left( {\sin x} \right)}^3}}}{{\sin x{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{{{{\left( {\cos x} \right)}^4} + 2{{\left( {\sin x} \right)}^2}{{\left( {\cos x} \right)}^2} + {{\left( {\sin x} \right)}^4}}}{{\sin x{{\left( {\cos x} \right)}^3}}} = \hfill \\ \end{gathered}$

18. anonymous

$(\cos(x)^4 +2\sin(x)^2(cosx)^2+\sin(x)^4)/\sin(x)xcos(x)^3$

19. Michele_Laino

hint: the numerator is a perfect square

20. Michele_Laino

hint: $\Large {\left\{ {{{\left( {\sin x} \right)}^2} + {{\left( {\cos x} \right)}^2}} \right\}^2} = ...?$

21. anonymous

(sinx)^4+2(sinx)^2(cosx)^2+(cosx)^2

22. anonymous

(cosx)^4

23. anonymous

((sinx^2 + cosx^2)^2)/sinx(cosx)^3

24. Michele_Laino

I got this: since $\Large \begin{gathered} {\left( {\cos x} \right)^4} + 2{\left( {\sin x} \right)^2}{\left( {\cos x} \right)^2} + \left( {\sin x} \right) = \hfill \\ = {\left\{ {{{\left( {\sin x} \right)}^2} + {{\left( {\cos x} \right)}^2}} \right\}^2} = {1^2} = 1 \hfill \\ \end{gathered}$ we have: $\large \begin{gathered} \cot x + 2\tan x + {\left( {\tan x} \right)^3} = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + {\left( {\frac{{\sin x}}{{\cos x}}} \right)^3} = \hfill \\ \hfill \\ = \frac{{\cos x}}{{\sin x}} + 2\frac{{\sin x}}{{\cos x}} + \frac{{{{\left( {\sin x} \right)}^3}}}{{{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{{\cos x{{\left( {\cos x} \right)}^3} + 2\sin x\sin x{{\left( {\cos x} \right)}^2} + \sin x{{\left( {\sin x} \right)}^3}}}{{\sin x{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{{{{\left( {\cos x} \right)}^4} + 2{{\left( {\sin x} \right)}^2}{{\left( {\cos x} \right)}^2} + {{\left( {\sin x} \right)}^4}}}{{\sin x{{\left( {\cos x} \right)}^3}}} = \hfill \\ \hfill \\ = \frac{1}{{\sin x{{\left( {\cos x} \right)}^3}}} \hfill \\ \hfill \\ \end{gathered}$

25. anonymous

oh

26. anonymous

ok I understand thanks

27. Michele_Laino

sorry, another typo... $\Large \begin{gathered} {\left( {\cos x} \right)^4} + 2{\left( {\sin x} \right)^2}{\left( {\cos x} \right)^2} + {\left( {\sin x} \right)^4} = \hfill \\ = {\left\{ {{{\left( {\sin x} \right)}^2} + {{\left( {\cos x} \right)}^2}} \right\}^2} = {1^2} = 1 \hfill \\ \end{gathered}$

28. anonymous

ok thanks

29. Michele_Laino

:)