Question

If tan 35° = k then find the value of
(tan 145° - tan 125°)/(1+tan145°.tan125°)

Answer 1

$\textbf{Given:}$

$\mathsf{tan\,35^\circ=k}$

$\textbf{To find:}$

$\textsf{The value of}\;\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}}$

$\textbf{Solution:}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}}$

$\mathsf{=\dfrac{tan(180^\circ-35^\circ)-tan(90^\circ+35^\circ)}{1+tan(180^\circ-35^\circ)\;tan(90^\circ+35^\circ)}}$

$\mathsf{=\dfrac{-tan35^\circ+cot35^\circ}{1+tan35^\circ\;cot35^\circ}}$

$\mathsf{=\dfrac{-tan35^\circ+\dfrac{1}{tan35^\circ}}{1+tan35^\circ(\dfrac{1}{tan35^\circ})}}$

$\mathsf{=\dfrac{-k+\dfrac{1}{k}}{1+1}}$

$\mathsf{=\dfrac{\dfrac{1-k^2}{k}}{2}}$

$\mathsf{=\dfrac{1-k^2}{2k}}$

$\implies\boxed{\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}=\dfrac{1-k^2}{2k}}}$

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Answer 2

Answer:

an35

∘

=k

\textbf{To find:}To find:

\textsf{The value of}\;\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}}The value of

1+tan145

∘

tan125

∘

tan145

∘

−tan125

∘

\textbf{Solution:}Solution:

\mathsf{Consider,}Consider,

\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}}

1+tan145

∘

tan125

∘

tan145

∘

−tan125

∘

\mathsf{=\dfrac{tan(180^\circ-35^\circ)-tan(90^\circ+35^\circ)}{1+tan(180^\circ-35^\circ)\;tan(90^\circ+35^\circ)}}=

1+tan(180

∘

−35

∘

)tan(90

∘

+35

∘

)

tan(180

∘

−35

∘

)−tan(90

∘

+35

∘

)

\mathsf{=\dfrac{-tan35^\circ+cot35^\circ}{1+tan35^\circ\;cot35^\circ}}=

1+tan35

∘

cot35

∘

−tan35

∘

+cot35

∘

\mathsf{=\dfrac{-tan35^\circ+\dfrac{1}{tan35^\circ}}{1+tan35^\circ(\dfrac{1}{tan35^\circ})}}=

1+tan35

∘

(

tan35

∘

1

)

−tan35

∘

+

tan35

∘

1

\mathsf{=\dfrac{-k+\dfrac{1}{k}}{1+1}}=

1+1

−k+

k

1

\mathsf{=\dfrac{\dfrac{1-k^2}{k}}{2}}=

2

k

1−k

2

\mathsf{=\dfrac{1-k^2}{2k}}=

2k

1−k

2

\implies\boxed{\mathsf{\dfrac{tan145^\circ-tan125^\circ}{1+tan145^\circ\;tan125^\circ}=\dfrac{1-k^2}{2k}}}⟹

1+tan145

∘

tan125

∘

tan145

∘

−tan125

∘

=

2k

1−k

2