Question

evaluate Sin48 sin42-cos48 cos42​

Answer 1

EVALUATE.

$\sf \: Cos48 \: x \: cos 42 - Sin 48 \: x \: sin 42$

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Formulas used

$\sf \: cos (90°-A) => Sin A$

$\sf \: sin (90°-A) = cos A$

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Solution..-

$\begin{gathered} \sf \: \cos(48) \times \cos(42) - \sin(48) \times \sin(42) \\ \\ \sf \implies \frac{ \cos(48) }{ \sin(42) } - \frac{ \sin(48) }{ \cos(42) } \\ \\ \sf \implies\frac{ \cos(48) }{ \ \sin(90 - 42) } - \frac{ \sin(48) }{ \cos(90 - 48) } \\ \\ \sf \implies \: \frac{ \cos(48) }{ \cos(48) } - \frac{ \sin(48) }{ \sin(48) } \\ \\ \sf \implies \bold{1 - 1 }\\ \\ \sf \implies \bold{ 0}\end{gathered}</p><p>$

Hence proved

Answer 2

$\large\underline{\sf{Solution-}}$

Given Trigonometric expression is

$\rm :\longmapsto\:sin48\degree sin42\degree - cos48\degree cos42\degree$

We know

$\boxed{ \tt{ \: 42\degree + 48\degree = 90\degree \rm \implies\:48\degree = 90\degree - 42\degree \: }}$

So, on substituting the value of 48°, we get

$\rm \: = \: sin(90\degree - 42\degree ) \: sin42\degree - cos(90\degree - 42\degree ) \: cos42\degree$

We know,

$\boxed{ \tt{ \: sin(90\degree - x) = cosx \: }} \\ \\ \bf \: and \\ \\ \boxed{ \tt{ \: cos(90\degree - x) = sinx \: }}$

So, on substituting these values, we get

$\rm \: = \: cos42\degree sin42\degree - cos42\degree sin42\degree$

$\rm \: = \: 0$

Hence,

$\rm :\longmapsto\:\boxed{ \tt{ \: sin48\degree sin42\degree - cos48\degree cos42\degree = 0 \: }}$

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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