7.05 Trigonometric identities
Express \tan x in terms of \cos x and \sin x .
Find the exact value of \tan 10 \degree - \dfrac{\sin 10 \degree}{\cos 10 \degree}.
Find the value of \cos x given the following:
\sin x = - \dfrac{\sqrt{3}}{2} and \tan x = -\sqrt{3}
\sin x = \dfrac{b}{c} and \tan x = \dfrac{b}{a}
Find the value of \sin x given the following:
\cos x = \dfrac{15}{17} and \tan x = \dfrac{8}{15}
\cos x = - \dfrac{5}{13} and \tan x \gt 0
Find the value of \tan x given the following:
Simplify the following expressions:
\tan \theta \cos \theta
\sin \left(90 \degree - y\right) \tan y
\dfrac{\sin \theta - \cos \theta}{\cos \theta}
\dfrac{\tan \theta}{\text{cosec } \theta \sec \theta}
Find the exact value of \tan x given the following equations:
State whether the following statements are correct:
\sin ^{2}\theta + \cos ^{2}\theta = 1
\tan ^{2}x + 1 = \sec ^{2}x
\sin ^{2}\theta + \cos ^{2}\theta = 2
1 + \cot ^{2}\theta = \sec ^{2}\theta
Find the exact value of the following:
Simplify the following expressions:
\left(\cos \theta - 1\right) \left(\cos \theta + 1\right)
\left(\cos \theta - \sin \theta\right)^{2}
\cos \theta \sin ^{2}\left(\theta\right) - \cos \theta
\left(3 - \cos x\right)^{2} + \sin ^{2}\left(x\right)
\left(1 + \tan ^{2}\left(u\right)\right) \left(1 - \sin ^{2}\left(u\right)\right)
\left(1 - \sec ^{2}\left(\theta\right)\right) \cot \theta
\text{cosec } ^{2}\left(\theta\right) - \cot ^{2}\left(\theta\right)
\cos \theta \left(\sec \theta - \cos \theta\right)
\left(1 - \text{cosec } \theta\right) \left(1 + \sin \theta\right)
\sec ^{2}\left(x\right) \left(\cos ^{2}\left(x\right) - 1\right)
\dfrac{1}{1 - \cos \theta} + \dfrac{1}{1 + \cos \theta}
\dfrac{\sin ^{2}\left(\theta\right)}{1 - \sin ^{2}\left(\theta\right)}
\dfrac{1}{1 - \sin x} \times \dfrac{1}{1 + \sin x}
\dfrac{1}{1 - \cos ^{2}\left(x\right)} - 1
\dfrac{1 - \sin ^{2}\left(\theta\right)}{\sin ^{2}\left(\theta\right) + \cos ^{2}\left(\theta\right)}
\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta}
\dfrac{1}{1 + \tan ^{2}\left(x\right)}
\tan \theta + \dfrac{1}{\tan \theta}
\dfrac{\sec ^{2}\left(x\right) - 1}{\text{cosec } ^{2}\left(x\right) - 1}
\dfrac{1}{\text{cosec } ^{2}\left(\theta\right)} + \dfrac{\cos \theta}{\sec \theta}
\dfrac{1}{\text{cosec } ^{2}\left(\theta\right) - 1}
\dfrac{1}{\cos ^{2}\left(x\right)} - \dfrac{1}{\cot ^{2}\left(x\right)}
\left(\sec \theta - \text{cosec } \theta\right) \left(\cos \theta + \sin \theta\right)
Simplify \sqrt{a^{2} + x^{2}}, where x = a \tan \theta, a is a constant, and 0 \degree \lt \theta \lt 90 \degree.
If x = 4 \sin \theta and y = 3 \cos \theta, form an equation relating x and y that does not involve \sin \theta or \cos \theta.
State the values of x that are not in the domain of the identify 1 + \cot ^{2}\left(x\right) = \text{cosec } ^{2}\left(x\right)
If \sin \theta = x, express \dfrac{1 - \cos ^{2}\left(\theta\right)}{\sec ^{2}\left(\theta\right)} in terms of x.
Prove the following identities:
\dfrac{\cos x \tan x}{\sin x} = 1
\dfrac{\sec \theta}{\tan \theta} = \text{cosec } \theta
\dfrac{1 - \sin ^{2}\left(x\right)}{\cos x} = \cos x
\dfrac{\sin \theta}{1 - \cos \theta} = \dfrac{1 + \cos \theta}{\sin \theta}
\dfrac{1 - \cot x}{1 + \cot x} = \dfrac{\tan x - 1}{\tan x + 1}
\left(\sin x + \cos x\right)^{2} = 1 + 2 \sin x \cos x
\sin A \cos A \tan A = \sin ^{2}\left(A\right)
\cos ^{4}\left(x\right) - \sin ^{4}\left(x\right) = 2 \cos ^{2}\left(x\right) - 1
5 \cos ^{2}\left(\theta\right) - 3 = 2 - 5 \sin ^{2}\left(\theta\right)
\dfrac{\left(1 + \sin \theta\right)^{2} + \cos ^{2}\left(\theta\right)}{1 + \sin \theta} = 2
\dfrac{\sin ^{2}\left(x\right) + \sin x \cos x}{\cos ^{2}\left(x\right) + \sin x \cos x} = \tan x
\dfrac{\sin x \cos \left(90 \degree - x\right)}{\cos x \sin \left(90 \degree - x\right)} = \tan ^{2}\left(x\right)
\tan \theta \sin \theta + \cos \theta = \sec \theta
\dfrac{\tan ^{2}\left(x\right) - 1}{\tan ^{2}\left(x\right) + 1} = 1 - 2 \cos ^{2}\left(x\right)
\dfrac{1 + \cot ^{2}\left(x\right)}{\text{cosec } x} = \text{cosec } x
\dfrac{\cos ^{2}\left(x\right)}{\sin x} = \text{cosec } x - \sin x
\tan x + \cot x = \text{cosec } x \sec x
\dfrac{\text{cosec } ^{4}\left(x\right) - \cot ^{4}\left(x\right)}{\text{cosec } ^{2}\left(x\right) + \cot ^{2}\left(x\right)} = 1
\dfrac{4 + \tan ^{2}\left(x\right) - \sec ^{2}\left(x\right)}{\text{cosec } ^{2}\left(x\right)} = 3 \sin ^{2}\left(x\right)
\dfrac{\cos \alpha}{1 + \sin \alpha} = \sec \alpha \left(1 - \sin \alpha\right)
\left(\sec x - \tan x\right)^{2} = \dfrac{1 - \sin x}{1 + \sin x}
\dfrac{\sin x}{\text{cosec } x - 1} = \dfrac{1 + \sin x}{\cot x}
Prove the following identities:
\tan ^{2}\left(y\right) - \sin ^{2}\left(y\right) = \tan ^{2}\left(y\right) \sin ^{2}\left(y\right)
\sin ^{2}\left(a\right) - \sin ^{2}\left(b\right) + \cos ^{2}\left(a\right) \sin ^{2}\left(b\right) - \sin ^{2}\left(a\right) \cos ^{2}\left(b\right) = 0
\left(1 - \sin ^{2}\left(x\right)\right) \left(1 + \sin ^{2}\left(x\right)\right) = 2 \cos ^{2}\left(x\right) - \cos ^{4}\left(x\right)
\sin \theta \left(1 + \tan \theta\right) + \cos \theta \left(1 + \cot \theta\right) = \dfrac{\sin \theta + \cos \theta}{\sin \theta \cos \theta}
A searchlight at the grand opening of a new car dealership casts a spot of light on a wall located 75 meters from the searchlight. The acceleration a of the spot of light is found to be a = 1200 \sec \theta \left( 2 \sec ^{2}\left(\theta\right) - 1\right). Show that this is equivalent to a = 1200 \left(\dfrac{1 + \sin ^{2}\left(\theta\right)}{\cos ^{3}\left(\theta\right)}\right).