Question

which of these are equivalent to 2tanx(sec^2x-1)/cos^3x​

Answer 1

Answer:

2 tan x( sec²x-1)/cos³x

WE KNOW THAT tan²theta=sec² theta-1

Hence,

2tan x(sec²x-1)= 2tan ( tan²x) = 2 tan³x

WE KNOW THAT cos theta=1/sec theta

Hence, cos³x=1/sec³x

so the fraction becomes:

2tan³x/1/sec³x

=2tan³x× sec³x

THEREFORE THE ANSWER IS ( 2tan³x)(sec³x)

Answer 2

To find:

The simplified value of $\mathsf{\dfrac{2\:tanx\:(sec^{2}x-1)}{cos^{3}x}}$

Trigonometric formulae:

Before we solve the problem, let us know some trigonometric formulae,

• $\mathsf{cosA\:secA=1}$

• $\mathsf{sec^{2}A-tan^{2}A=1}$

Step-by-step explanation:

Now, $\mathsf{\dfrac{2\:tanx\:(sec^{2}x-1)}{cos^{3}x}}$

$\mathsf{=\dfrac{2\:tanx\:tan^{2}x}{cos^{3}x}}$

• Formula used: $\mathsf{sec^{2}A-1=tan^{2}A}$

$\mathsf{=2\:tan^{3}x\times (\dfrac{1}{cosx})^{3}}$

$\mathsf{=2\:tan^{3}x\:(secx)^{3}}$

• Formula used: $\mathsf{\dfrac{1}{cosA}=secA}$

$\mathsf{=2\:tan^{3}x\:sec^{3}x}$

Final Answer:

The simplified form of $\mathsf{\dfrac{2\:tanx\:(sec^{2}x-1)}{cos^{3}x}}$ is $\mathsf{2\:tan^{3}x\:sec^{3}x}$.