Evaluate cos-sin 90 - - - sin - -sin 180 - - - tan 90 - -

Consider the given #10;expression #10; #10; cosθsin #10;( 90^∘+θ #160; )+sin #10;( θ #160; )sin( #10;180^∘+θ #160; )+tan #10;( 90^∘+θ ...

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Standard X

Mathematics

Trigonometric Ratios of Complementary Angles

Evaluate co...

Question

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

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Solution

Consider the given expression

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

We know that

$sin (90^{\circ} + θ) = cos θ$

$sin (- θ) = - sin θ$

$sin (180^{\circ} + θ) = - sin θ$

$tan (90^{\circ} + θ) = - cot θ$

Therefore,

$= \frac{cos θ}{cos θ} + \frac{- sin θ}{- sin θ} - \frac{cot θ}{cot θ}$

$= 1 + 1 - 1$

$= 1$

Hence, the value is $1$ .

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Similar questions

Q. The value of

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

Q. The value of :

\frac{c o s θ}{s i n (90 + θ)} + \frac{s i n (- θ)}{s i n (180 + θ)} + \frac{t a n (90 + θ)}{c o t θ}

Q. Simplify

\frac{sec (90^{\circ} - θ)}{sin (90^{\circ} - θ)} \times \frac{cos θ}{tan (90^{\circ} - θ)} - sec θ

Prove that:

(i) $s i n θ c o s (90^{\circ} - θ) + s i n (90^{\circ} - θ) c o s θ = 1$

(ii) $\frac{s i n θ}{c o s (90^{\circ} - θ)} + \frac{c o s θ}{s i n (90^{\circ} - θ)} = 2$

(iii) $\frac{s i n θ c o s (90^{\circ} - θ) c o s θ}{s i n (90^{\circ} - θ)} + \frac{c o s θ s i n (90^{\circ} - θ) s i n θ}{c o s (90^{\circ} - θ)} = 1$

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

(v) $\frac{c o s (90^{\circ} - θ)}{1 + s i n (90^{\circ} - θ)} + \frac{1 + s i n (90^{\circ} - θ)}{c o s (90^{\circ} - θ)} = 2 c o s e c θ$

(vi) $\frac{s e c (90^{\circ} - θ) c o s e c θ - t a n (90^{\circ} - θ) c o t θ + c o s^{2} 25^{\circ} + c o s^{2} 65^{\circ}}{3 t a n 27^{\circ} t a n 63^{\circ}} = \frac{2}{3}$

(vii) $c o t θ t a n (90^{\circ} - θ) - s e c (90^{\circ} - θ) c o s e c θ + \sqrt{3} t a n 12^{\circ} t a n 60^{\circ} t a n 78^{\circ} = 2$

Q. Evaluate:

\frac{cos θ}{sin (90^{\circ} - θ)} + \frac{sin θ}{cos (90^{\circ} - θ)}

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Evaluate cosθsin(90+θ)+sin(−θ)sin(180+θ)+tan(90+θ)cotθ

Consider the given expression cosθsin(90∘+θ)+sin(−θ)sin(180∘+θ)+tan(90∘+θ)cotθ We know that sin(90∘+θ)=cosθ sin(−θ)=−sinθ sin(180∘+θ)=−sinθ tan(90∘+θ)=−cotθ Therefore, =cosθcosθ+−sinθ−sinθ−cotθcotθ =1+1−1 =1 Hence, the value is 1.

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