Simplify cos 2 180 - - - cos 2 27 - - - -sin 180 - - sin - - - cos 2 27 - - - sin 180 - -

cos^2(180 θ)+ #10;cos^2(270+θ)sin (180+θ)sin ( θ)+ #10;cos^2(270+θ)sin (180+θ) #10; #10; =cos^2+sin^2θ sin #10;θ sinθ+ sin^2θ sinθ #10; ...

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Byju's Answer

Standard X

Mathematics

Trigonometric Identities

simplify cos...

Question

simplify
$\frac{{cos}^{2} (180 - θ) + \frac{{cos}^{2} (27 θ + θ)}{sin (180 + θ)}}{sin (- θ)} + \frac{{cos}^{2} (27 θ + θ)}{sin (180 + θ)}$

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Solution

$\frac{{cos}^{2} (180 - θ) + \frac{{cos}^{2} (270 + θ)}{sin (180 + θ)}}{sin (- θ)} + \frac{{cos}^{2} (270 + θ)}{sin (180 + θ)}$

$= \frac{{cos}^{2} + \frac{{sin}^{2} θ}{- sin θ}}{- sin θ} + \frac{{sin}^{2} θ}{- sin θ}$

$= \frac{{cos}^{2} θ - sin θ + {sin}^{2} θ}{- sin θ}$

$= \frac{1 - sin θ}{- sin θ} = 1 - csc θ$

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Similar questions

Q. Prove that
(i)

\frac{\cos (2 π + θ) cosec (2 π + θ) \tan (π / 2 + θ)}{\sec (π / 2 + θ) cosθ \cot (π + θ)} = 1

(ii)

\frac{cosec (90 ° + θ) + \cot (450 ° + θ)}{cosec (90 ° - θ) + \tan (180 ° - θ)} + \frac{\tan (180 ° + θ) + \sec (180 ° - θ)}{\tan (360 ° + θ) - \sec (- θ)} = 2

(iii)

\frac{\sin (180 ° + θ) \cos (90 ° + θ) \tan (270 ° - θ) \cot (360 ° - θ)}{\sin (360 ° - θ) \cos (360 ° + θ) cosec (- θ) \sin (270 ° + θ)} = 1

(iv)

\{1 + cotθ - \sec (\frac{π}{2} + θ)\} \{1 + cotθ + \sec (\frac{π}{2} + θ)\} = 2 cotθ

(v)

\frac{\tan (90 ° - θ) \sec (180 ° - θ) \sin (- θ)}{\sin (180 ° + θ) \cot (360 ° - θ) cosec (90 ° - θ)} = 1

Prove that:

$(i) \frac{c o s (2 π + θ) c o s e c (2 π + θ) t a n (π / 2 + θ)}{s e c (π / 2 + θ) c o s θ c o t (π + θ)} = 1$

$(i i) \frac{c o s e c (90^{\circ} + θ) + c o t (450^{\circ} + θ)}{c o s e c (90^{\circ} - θ) + t a n (180^{\circ} - θ)} + \frac{t a n (180^{\circ} + θ) + s e c (180^{\circ} - θ)}{t a n (360^{\circ} + θ) - s e c (- θ)} = 2$

$(i i i) \frac{s i n (180^{\circ} + θ) c o s (90^{\circ} + θ) t a n (270^{\circ} - θ) c o t (360^{\circ} - θ)}{s i n (360^{\circ} - θ) c o s (360^{\circ} + θ) c o s e c (- θ) s i n (270^{\circ} + θ)} (i v) 1 + c o t θ - s e c (\frac{π}{2} + θ)} 1 + c o t θ + s e c (\frac{π}{2} + θ)} = 2 c o t θ$

$(v) \frac{t a n (90^{\circ} - θ) s e c (180^{\circ} - θ) s i n (- θ)}{s i n (180^{\circ} + θ) c o t (360^{\circ} - θ) c o s e c (90^{\circ} - θ)} = 1$

Q. Evaluate

\frac{cos θ}{sin (90 + θ)} + \frac{sin (- θ)}{sin (180 + θ)} + \frac{tan (90 + θ)}{cot θ}

Q. Find

a

\frac{cos (90^{\circ} + θ) sec (- θ) tan (180^{\circ} - θ)}{sec (360^{\circ} - θ) sin (180^{\circ} + θ) cot (90^{\circ} - θ)} = - a

Q. Solve:

\frac{cos (90 + θ) sec (- θ) tan (180^{\circ} - θ)}{sec (360^{\circ} - θ) sin (180^{\circ} + θ) cot (90^{\circ} - θ)} =

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simplifycos2(180−θ)+cos2(27θ+θ)sin(180+θ)sin(−θ)+cos2(27θ+θ)sin(180+θ)

cos2(180−θ)+cos2(270+θ)sin(180+θ)sin(−θ)+cos2(270+θ)sin(180+θ) =cos2+sin2θ−sinθ−sinθ+sin2θ−sinθ =cos2θ−sinθ+sin2θ−sinθ =1−sinθ−sinθ=1−cscθ

simplify
$\frac{{cos}^{2} (180 - θ) + \frac{{cos}^{2} (27 θ + θ)}{sin (180 + θ)}}{sin (- θ)} + \frac{{cos}^{2} (27 θ + θ)}{sin (180 + θ)}$