Question

If tan5Θ = cot4Θ, Find cos3Θ.​

Answer 1

$\LARGE{ \underline{\underline{ \sf{\red{Required \: answer:}}}}}$

GiveN:

• tan 5Θ = cot 4Θ

We have to FinD:

• Value of cos 3Θ ?

Step-wise-Step Explanation:

We have,

➛ tan 5Θ = cot 4Θ

This can be written as,

➛ tan 5Θ = tan (90° - 4Θ)

Then,

➛ 5Θ = 90° - 4Θ

➛ 5Θ + 4Θ = 90°

➛ 9Θ = 90°

➛ Θ = 10°

We need to find: cos 3Θ

Plugging the value of Θ

= cos (3 × 10°)

= cos 30°

= √3 / 2

Hence,

• The required value of cos 3Θ is √3/2.

Extra Tips:

For questions of complementary angles in trigonometry always have a idea about the relationship between the trigonometric ratios. Like:

• sin (90°- A) = cos A ;
• cos (90°- A) = sin A

• tan (90°- A) = cot A;
• cot (90°- A) = tan A

• sec (90°- A) = csc A;
• cosec (90°- A) = sec A

Answer 2

$\underline{\underline{\sf{\clubsuit \:\:Question}}}$

• If tan5θ  = cot4θ , Find cos3θ

$\underline{\underline{\sf{\clubsuit \:\:Given}}}$

• tan5θ  = cot4θ

$\underline{\underline{\sf{\clubsuit \:\:To\:\:Find}}}$

• cos3θ

$\underline{\underline{\sf{\clubsuit \:\:Answer}}}$

• cos3θ = √3 / 2

$\underline{\underline{\sf{\clubsuit \:\:Calculations}}}$

• We know cot4θ = tan (90° - 4θ)

But How ?

cot4θ = tan (90° - 4θ)

Manipulating right side

tan (90° - 4θ)

Solve tan (90° - 4θ)

$\sf{\text { Use the basic trigonometric identity: } \tan (x)=\dfrac{\sin (x)}{\cos (x)}}$

$\sf{=\dfrac{\sin \left(90^{\circ \:}-4\theta\right)}{\cos \left(90^{\circ \:}-4\theta\right)}}$

Use the Angle Difference identity: sin(s - t) = sin(s)cos(t) - cos(s)sin(t)

$\sf{=\dfrac{\sin \left(90^{\circ \:}\right)\cos \left(4\theta\right)-\cos \left(90^{\circ \:}\right)\sin \left(4\theta\right)}{\cos \left(90^{\circ \:}-4\theta\right)}}$

Use the Angle Difference identity: cos(s - t) = cos(s)cos(t) + sin(s)sin(t)

$\sf{=\dfrac{\sin \left(90^{\circ \:}\right)\cos \left(4\theta\right)-\cos \left(90^{\circ \:}\right)\sin \left(4\theta\right)}{\cos \left(90^{\circ \:}\right)\cos \left(4\theta\right)+\sin \left(90^{\circ \:}\right)\sin \left(4\theta\right)}}$

$\sf{=\dfrac{\cos \left(4\theta\right)}{\cos \left(90^{\circ \:}\right)\cos \left(4\theta\right)+\sin \left(90^{\circ \:}\right)\sin \left(4\theta\right)}}$

$\sf{=\dfrac{\cos \left(4\theta\right)}{\sin \left(4\theta\right)}}$

$\sf{\mathrm{Use\:the\:following\:identity:}\:\dfrac{\cos \left(x\right)}{\sin \left(x\right)}=\cot \left(x\right)}$

$\sf{=\cot \left(4\theta\right)}$

Hence cot4θ = tan (90° - 4θ)

We have :

tan5θ  = cot4θ

⇒ tan5θ  = tan (90° - 4θ)

Take off tan :

⇒ 5θ  = 90° - 4θ

Add 4θ on both sides :

⇒ 5θ + 4θ  = 90° - 4θ + 4θ

⇒ 5θ + 4θ  = 90°

⇒ 9θ  = 90°

Divide both sides by 9 :

⇒ 9θ/9  = 90°/9

⇒ θ = 10°

Now, Find the value of cos3θ

Substitute the value of θ :

cos3θ

= cos (3 × 10°)

= cos 30°

= √3 / 2

∴ cos3θ = √3 / 2