Question

tan 25° tan 65° - cot 25° cot 65°.
please answer with step by step explanation...​

Answer 1

$\begin{gathered}\Large{\bold{\blue{\underline{Formula \: Used \::}}}} \end{gathered}$

$(1). \: \boxed{ \green{ \bf \:tan(90 - x) = cotx }}$

$(2). \: \boxed{ \green{ \bf \:tanx \: \times \: cotx \: = \: 1 }}$

$\large\underline\purple{\bold{Solution :- }}$

$\longmapsto\tt \: tan 25\degree \:tan65\degree \: - cot25\degree \:cot65\degree \:$

$\tt \longmapsto\: tan25\degree \:tan(90\degree \: - 25\degree \:) - cot25\degree \:cot(90\degree \: - 25\degree \:)$

$\tt \longmapsto\: \cancel{tan25\degree \:cot25\degree \:} - \cancel{cot25\degree \:tan25\degree \:}$

$\rm :\implies\:0$

$\boxed{ \green{ \bf \: \longmapsto\: tan 25\degree \:tan65\degree \: - cot25\degree \:cot65\degree \: = 0}}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1