Find sin 80 – cos 70
a) cos 50
b) cos 40
c) cos 75
d) cos 60
Answers
Answer:
Step-by-step explanation:
We have to prove that sin80^{{\circ}}-cos70^{{\circ}}=cos50^{{\circ}}sin80
∘
−cos70
∘
=cos50
∘
⇒sin80^{{\circ}}=cos50^{{\circ}}+cos70^{{\circ}}sin80
∘
=cos50
∘
+cos70
∘
Now, using formula, cos(C+D)= 2cos\frac{C+D}{2}cos\frac{C-D}{2}cos(C+D)=2cos
2
C+D
cos
2
C−D
⇒sin80^{{\circ}}=2 cos\frac{50+70}{2}cos\frac{50-70}{2}sin80
∘
=2cos
2
50+70
cos
2
50−70
⇒sin80^{{\circ}}=2cos60^{{\circ}}cos(-10)^{{\circ}}sin80
∘
=2cos60
∘
cos(−10)
∘
We know that,cos(-\alpha)=cos\alphacos(−α)=cosα ,therefore,
⇒sin80^{{\circ}}=2cos60^{{\circ}}cos10^{{\circ}}sin80
∘
=2cos60
∘
cos10
∘
⇒sin80^{{\circ}}=2{\times}\frac{1}{2}{\times}cos(90^{{\circ}}-80^{{\circ}})sin80
∘
=2×
2
1
×cos(90
∘
−80
∘
)
⇒sin80^{{\circ}}=sin80^{{\circ}}sin80
∘
=sin80
∘
Since, L.H.S=R.H.S, hence proved.
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Answer:
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