Prove that - 3 -2- sec -5 -2-tan5 -2- tan-3 -2-1

LHS =sec ( 3 π/2 θ ) sec ( θ 5 π/2 )+tan ( 5 π/2+θ )tan ( θ 3 π/2 ) =sec ( 3 π/2 θ )sec ( ( 5 π/2 θ ) )+tan ( 5 π/2 θ )tan [ ( 3 π/2 θ ) ] = cosec ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Allied Angles

Prove that : ...

Question

Prove that : $s e c (\frac{3 π}{2} - θ) s e c (θ - \frac{5 π}{2}) + t a n (\frac{5 π}{2} + θ) t a n (θ - \frac{3 π}{2}) = - 1$

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Solution

$L H S = s e c (\frac{3 π}{2} - θ) s e c (θ - \frac{5 π}{2}) + t a n (\frac{5 π}{2} + θ) t a n (θ - \frac{3 π}{2})$

$= s e c (\frac{3 π}{2} - θ) s e c (- (\frac{5 π}{2} - θ)) + t a n (\frac{5 π}{2} - θ) t a n [- (\frac{3 π}{2} - θ)]$

$= - c o s e c θ . s e c (\frac{5 π}{2} - θ) = c o t θ \times (-) t a n (\frac{3 π}{2} - θ)$

$⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} ∵ (s e c (\frac{3 π}{2} - θ)) = - c o s e c θ, s e c (- θ) = s e c θ, t a n (\frac{5 π}{2} + θ) = - c o t θ a n d t a n (- θ) = - (t a n θ) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$= - c o s e c θ \times c o s e c θ - c o t θ \times (- 1) \times c o t θ$

$⎡ ⎢ ⎣ \begin{matrix} ∵ (s e c (\frac{5 π}{2} - θ)) = - c o s e c θ a n d t a n (\frac{3 π}{2} - θ) = c o t θ \end{matrix} ⎤ ⎥ ⎦$

$= - c o s e c^{2} θ \times c o t^{2} θ$

$= - c o s e c^{2} θ \times c o s e c^{2} θ - 1 [∵ c o s e c^{2} θ = 1 + c o t^{2} θ]$

=-1 =RHS Hence proved.

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Similar questions

Q. Show that

sec (\frac{3 π}{2} - θ) sec (θ - \frac{5 π}{2}) + tan (\frac{5 π}{2} + θ) tan (θ - \frac{3 π}{2}) = - 1

Q. Prove that

cos (\frac{3 π}{2} + θ) cos (2 π + θ) [cot (\frac{3 π}{2} - θ) + cot (2 π + θ)] = 1

Q. Let

A = {θ : 2 {cos}^{2} + θ + sin θ \leq 2}

and

B = {θ : π / 2 \leq θ \leq 3 π / 2} .

Then

A \cap B = {θ : π / 2 \leq θ \leq 5 π / 6 o r π \leq θ \leq 3 π / 4}

If true enter 1 else 0

Q. Solve

2 {cos}^{2} θ + sin θ \leq 2

where

π / 2 \leq θ \leq 3 π / 2

Q. If

sec θ = \sqrt{2}

and

\frac{3 π}{2} < θ < 2 π

, if value of

\frac{1 + tan θ + csc θ}{1 + cot θ - csc θ} = m

then find

- m

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Prove that : sec(3π2−θ)sec(θ−5π2)+tan(5π2+θ)tan(θ−3π2)=−1

Prove that : $s e c (\frac{3 π}{2} - θ) s e c (θ - \frac{5 π}{2}) + t a n (\frac{5 π}{2} + θ) t a n (θ - \frac{3 π}{2}) = - 1$