Prove that: (i) sin θ cos (90∘−θ)+sin(90∘−θ)cos θ=1 (ii) s

1.	Prove that: (i) sin θ cos (90∘−θ)+sin(90∘−θ)cos θ=1 (ii) sin θcos (90∘−θ)+cos θsin (90∘−θ)=2 (iii) sin θ cos(90∘−θ)cos θsin (90∘−θ)+cos θ sin (90∘−θ)sin θcos (90∘−θ)=1 (iv) cos(90∘−θ)sec(90∘−θ)tan θcosec(90∘−θ)sin(90∘−θ)cot(90∘−θ)+tan(90∘−θ)cot θ=2 (v) cos(90∘−θ)1+sin(90∘−θ)+1+sin(90∘−θ)cos(90∘−θ)=2cosec θ (vi) sec(90∘−θ)cosec θ−tan(90∘−θ)cot θ+cos225∘+cos265∘3 tan 27∘ tan 63∘=23 (vii) cot θ tan(90∘−θ)−sec(90∘−θ)cosec θ+√3 tan 12∘ tan 60∘ tan 78∘=2
Answer» 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) $c o t θ t a n (90^{\circ} - θ) - s e c (90^{\circ} - θ) c o s e c θ + \sqrt{3} t a n 12^{\circ} t a n 60^{\circ} t a n 78^{\circ} = 2$

Prove that: (i) sin θ cos (90∘−θ)+sin(90∘−θ)cos θ=1 (ii) sin θcos (90∘−θ)+cos θsin (90∘−θ)=2 (iii) sin θ cos(90∘−θ)cos θsin (90∘−θ)+cos θ sin (90∘−θ)sin θcos (90∘−θ)=1 (iv) cos(90∘−θ)sec(90∘−θ)tan θcosec(90∘−θ)sin(90∘−θ)cot(90∘−θ)+tan(90∘−θ)cot θ=2 (v) cos(90∘−θ)1+sin(90∘−θ)+1+sin(90∘−θ)cos(90∘−θ)=2cosec θ (vi) sec(90∘−θ)cosec θ−tan(90∘−θ)cot θ+cos225∘+cos265∘3 tan 27∘ tan 63∘=23 (vii) cot θ tan(90∘−θ)−sec(90∘−θ)cosec θ+√3 tan 12∘ tan 60∘ tan 78∘=2

Answer»

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

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