Question

cot theta tan (90°- theta) - sec (90°- theta) cosec theta + ( sin square 25° + sin square 65°) + √3 ( tan 5° * tan 15° * tan 30° * tan 75° * tan 85°​

Answer 1

${\boxed{\mathtt{\green{To\:Find}}}}$

$Value \: of \: = \: \cot( \theta) . \: \tan(90 - \theta) - \: \sec(90 - \theta) . \: \csc( \theta) + ( { \sin(25) }^{2} + { \sin(65) }^{2} ) + \sqrt{3} ( \tan(5) . \: \tan(15) . \: \tan(30) . \: \tan(75) . \: \tan(85) )$

${\boxed{\mathtt{\green{Solution}}}}$

➾ Let's have a look on some important identities :-

$\sin(90 - \theta) = \cos( \theta)$

$\tan(90 - \theta) = \: \cot( \theta)$

$\sec(90 - \theta) = \: \csc( \theta)$

$\cos(90 - \theta) \: = \: \sin( \theta)$

$\cot(90 - \theta) \: = \: \tan( \theta)$

$\csc(90 - \theta) = \: \sec( \theta)$

➾ Now applying these identities in the question :-

$\implies \: \cot( \theta) . \cot( \theta) - \: \csc( \theta) . \: \csc( \theta)$

$+ ({ \sin(25) }^{2} \: + \: { \sin(90 - 25) }^{2} )$

$+ \sqrt{3} (( \tan(5) \times \tan(85) ).( \tan(15) \times \tan(75) ). \tan(30) )$

Some more rules :-

➾ (sin²a + cos²a = 1 )

➾ ( tan a × cot a = 1 )

➾ tan 30° = 1/√3

➾ Cot a =. Cos a/ Sin a

➾ Tan a = Sin a/ Cos a

➾ Cosec a = 1/ sin a

$\implies \: \frac{ \cot( \theta) }{ \tan( \theta) } + { \csc( \theta) }^{2} + ( { \sin(25) }^{2} + { \cos(25) }^{2})$

$+ \sqrt{3} ( (\tan(90 - 5) \times \tan(85) ).( \tan(90 - 75) \times \tan(75) ). \frac{1}{ \sqrt{3} } )$

$\implies \: \frac{ { \cos( \theta) }^{2} }{ { \sin( \theta) }^{2} } - \frac{1}{ { \sin( \theta) }^{2} } + 1 + \: \frac{ \sqrt{3} }{ \sqrt{3} } ( \cot(75) \times \tan(75) .( \cot(85) . \tan(85) )$

$\implies \: \frac{ { \cos( \theta) }^{2} - 1 }{ { \sin( \theta) }^{2} } + 1 + 1$

➾ cos²a - 1 = - sin²a

$\implies \: - \frac{ { \sin( \theta) }^{2} }{ { \sin( \theta) }^{2} } + 2$

$\implies \: - 1 + 2$

${\boxed{\mathtt{\green{= \: 1 }}}}$

Answer 2

Answer:

1.

Step-by-step explanation:

Cot ∅ * tan (90° - ∅) - Sec (90° - ∅) * Cosec ∅ + (Sin² 25 + Sin² 65) + √3 ( tan 5° * tan 15° * tan 30° *

tan 75° * tan 85°)

=> Cot ∅ * Cot ∅ - Cosec ∅ * Cosec ∅ + (Sin² (90-65)° + Sin² 65°)+ √3 [tan (90 - 85)° * tan (90 - 75)° * 1/√3 * tan 75° * tan 85°]

=> Cot² ∅ - Cosec² ∅ +( Cos² 65° + Sin² 65°) + (Cot 85° * Cot 75° * tan 75° * tan 85°)

=> Cos²∅*1/Sin² ∅ - 1/Sin² ∅ + 1 + (1/tan 85° * tan 85° * 1/tan 75° * tan 75°)

( Cot² ∅ = Cos² ∅/Sin² ∅ ; Cosec² ∅ = 1/Sin² ∅)

=> (Cos² ∅ - 1)/Sin² ∅+ 1 +1 (Tan 85° ; Tan 75° are cancelled out here).

( Cos² ∅ - 1 = - Sin² ∅)

=> - Sin² ∅ /Sin² ∅ + 2

=> - 1 + 2

=> 1.

Some trigonometric formulae:

• Cos² ∅ + Sin² ∅ = 1
• Sec² ∅ - tan² ∅ = 1
• Cosec² ∅ - Cot² ∅ = 1
• Sin (90° - ∅) = Cos ∅
• Cos (90° - ∅) = Sin ∅
• Tan (90° - ∅) = Cot ∅
• Cosec (90° - ∅) = Sec ∅
• Sec (90° - ∅) = Cosec ∅
• Cot (90° - ∅) = Tan ∅.