# MCQs on Trigonometry – Set 2

###### Explanation;

tan (πcosθ)=cot(πsinθ)

⇒sin(πcosθ)∕cos(πcosθ) = cos(πsinθ)∕sin(πsinθ)

⇒sin(πcosθ).sin(πsinθ)=cos(πsinθ).cos(πcosθ)

⇒sin(πcosθ).sin(πsinθ)-cos(πsinθ).cos(πcosθ)

⇒cos(πcosθ+πsinθ)=0

⇒πcosθ+πsinθ=±π∕2

cosθ+sinθ=±1∕2

Multiply both sides by 1∕√2

⇒ 1∕√2cosθ+1∕√2sinθ=±1∕2√2

since cosπ∕4=1∕√2=sinπ∕4

⇒cosπ∕4.cosθ+sinπ∕4.π∕4.sinθ=±1∕2√2

⇒cos(θ-π∕4)=±1∕2√2

###### Explanation;

Use componendo dividendo theorem

###### Explanation:

sin θ= sin α

⇒ sin θ – sin α=0

⇒ 2cos θ+α ∕2. sin θ-α ∕2

now one of these is 0 i-e

cos θ+α ∕2 =0 or sin θ-α ∕2=0

⇒ θ+α ∕2 is any odd multiple of π∕2 and θ-α∕2 is any multiple of π

###### Explanation

Given

α+β+γ=2π

⇒α∕2+β∕2+γ∕2=π

⇒tan (α∕2+β∕2)=tan(π-γ∕2)

⇒tanα∕2+tanβ∕2 ∕ 1-tanα∕2.tanβ∕2 = -tanγ∕2

⇒tanα∕2+tanβ∕2+tanγ∕2=tanα∕2.tanβ∕2.tanγ∕2

###### Explanation:

1+sin x + sin² x +sin ³ x +………..+∞ = -4+2√3

⇒1/ 1- sin x = -4+2√3

⇒sin x = √3/2

⇒ x = π/3 or 2π/3

###### Explanation:

You can prove it by putting values of m and n

###### Explanation

tan 31°.tan 32°.tan 33° ……………tan 59°

=(tan 31°. tan58°) X (tan 32°. tan 58°)………x(tan 44°. tan 46°)x (tan 45°)

=(tan 31° . tan(90°-31°)) x (tan 32°. tan (90°-32°)) x ………. x (tan 44° .tan(90°-44°)) x tan 45°

= (tan 31° cot 31°) x (tan 32° . cot 32°) x ……..x(tan44° . cot 44°) x 1 = 1

###### Explanation:

tan[-11π/6] . tan[21π/4] . cot [283π/6]

or

tan[-330] . tan[945] . cot [8490]

or

-tan[360-30] . tan[2×360+225] . cot [23×360+210]

or

tan[30] . tan[225] . cot [210]

or

tan[30] . tan[π+45] . cot [π+30]

or

tan[30] . tan[45] . cot [30]

or

1/√3.1.√3

which ar in G.P. with common ratio of √3

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