Question 513 Marks
Prove the following.
$\frac{\tan \theta}{\sec \theta+1}=\frac{\sec \theta-1}{\tan \theta}$
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If $\cot \theta=\frac{40}{9}$, find the values of cosecθ and sinθ.
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Prove the following.
sec6x – tan6x = 1 + 3 sec2x × tan2x
Question 543 Marks
If $\tan \theta=\frac{3}{4}$, find the values of secθ and cosθ.
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Prove the following.
sec2θ + cosec2θ = sec2θ × cosec2θ
Question 563 Marks
A tree was broken due to storm. Its broken upper part was so inclined that its top touched the ground making an angle of 30°with the ground.The distance from the foot of the tree and the point where the top touched the ground was 10 metre. What was the height of the tree.
Question 573 Marks
From the top of a building, an observer is looking at a scooter parked at some distance away, makes an angle of depression of $30^{\circ}$. If the height of the building is $40 \mathrm{~m}$, find how far the scooter is from the building. $(\sqrt{3}=1.73)$
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An observer at a distance of $10 \mathrm{~m}$ from a tree looks at the top of the tree, the angle of elevation is $60^{\circ}$. What is the height of the tree ? $(\sqrt{3}=1.73)$
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Eliminate $\theta$ from given equations.
x=a $\cot$ $\theta$ -b cosec $\theta$
y=a $\cot$ $\theta$+b cosec $\theta$
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If $5 \sin \theta-12 \cos \theta=0$, find the values of $\sec \theta$ and $\operatorname{cosec} \theta$..
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Prove : $\sin ^4 \theta-\cos ^4 \theta=1-2 \cos ^2 \theta$
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Prove : $\sec \theta+\tan \theta=\frac{1}{\sec \theta-\tan \theta}$
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Prove : $\operatorname{cosec}^2 A-\cos ^2 A=\frac{\sec ^2 A-\sin ^2 A}{\tan ^2 A}$
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Prove : $\left(\frac{1}{\cos \theta}+\frac{1}{\cot \theta}\right)=(\sec \theta-\tan \theta)=1$
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Prove : $\left(1+\tan ^2 \theta\right)(1+\sin \theta)(1-\sin \theta)=1$
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Prove : $\left(1+\cot ^2 \theta\right)(1+\cos \theta)(1-\cos \theta)=1$
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Prove : $\frac{\tan A+\sec A-1}{\tan A-\sec A+1}=\frac{1+\sin A}{\cos A}$
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Prove : $\frac{\tan ^3 A-1}{\tan A -1}=\sec ^2 A+\tan A$
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Prove : $\frac{\sin \theta+\tan \theta}{\cos \theta}=\tan \theta(1+\sec \theta)$
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Prove : $\frac{\cos \theta}{1+\sin \theta}=\sec \theta-\tan \theta$
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Prove : $\frac{\cos ^2 \theta}{1-\tan \theta}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}==1+\sin \theta$
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Prove : $\frac{\cos ^2 A+\tan ^2 A-1}{\sin ^2 A}=\tan ^2 A$
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Prove : $\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^2 \theta$
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Prove : $\cot ^2 \theta-\frac{1}{\sin ^2 \theta}=-1$
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Prove : $\cos ^2 \theta+\frac{1}{1+\cot ^2 \theta}=1$
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If tan $\theta$ = 2, find the values of other trigonometric ratios using the identities.
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If $\sqrt{3} \tan \theta=3 \sin \theta$, find the value of $\sin ^2 \theta-\cos ^2 \theta$.
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If cot $\theta$ = $\frac{7}{24}$, find the values of other trigonometric ratios using the identity.
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For a person standing at a distance of 80 m from a temple, the angle of elevation of its top is 45. Find the height of the church.
Question 803 Marks
$3 \sin \theta-4 \cos \theta=0$, then find the values of all trigonometric ratios.
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