How do you simplify #(cos^2 x)(tan x + cot x)#?

Question

Charles Autterson · Accepted Answer

Simplify #(cos^2 x)(tan x + cot x)#

Ans: cot x

#cos^2 x(sin x/cos x + cos x/sin x) = cos^2 x(1/(sin x.cos x))# = #= cos x/sin x = cot x#

Brooklyn Anderson · Answer

undefined To simplify $ (\cos^2 x)(\tan x + \cot x) $, use trigonometric identities:

1. Rewrite $ \tan x $ and $ \cot x $ in terms of $ \sin x $ and $ \cos x $ using the definitions of tangent and cotangent.
2. Substitute these expressions into the expression.
3. Use trigonometric identities to simplify the resulting expression.

Let's proceed:

1. Recall that $ \tan x = \frac{\sin x}{\cos x} $ and $ \cot x = \frac{\cos x}{\sin x} $.
2. Substitute:
   $$ (\cos^2 x)\left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right) $$
3. Now, simplify:
   $$ (\cos^2 x)\left(\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\right) $$
   Since $ \sin^2 x + \cos^2 x = 1 $:
   $$ (\cos^2 x)\left(\frac{1}{\sin x \cos x}\right) $$
   Further simplify using the identity $ \sin x \cos x = \frac{1}{2} \sin(2x) $:
   $$ \frac{\cos^2 x}{\frac{1}{2} \sin(2x)} $$
   Now, multiply the numerator and denominator by $ 2 $:
   $$ \frac{2 \cos^2 x}{\sin(2x)} $$

Therefore, $ (\cos^2 x)(\tan x + \cot x) $ simplifies to $ \frac{2 \cos^2 x}{\sin(2x)} $.