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Mathematics
Trigonometric transformation formulae

Prove that $tan {15^ \circ } + \tan {30^ \circ } + \tan {15^ \circ }\tan {30^ \circ } = 1$


Mathematics
Trigonometric transformation formulae
How do you verify the identity $ \sin \left( {\dfrac{\pi }{6} + x} \right) + \sin \left( {\dfrac{\pi }{6} - x} \right) = \cos x $ ?

Mathematics
Trigonometric transformation formulae
Identify transformations of trigonometric expressions prove the following identities $\dfrac{{{{\sin }^2}2a + 4{{\sin }^4}a - 4{{\sin }^2}a{{\cos }^2}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}} = {\tan ^4}a$

Mathematics
Trigonometric transformation formulae
If we have a trigonometric expression as $3\sin x+4\cos x=5$ , then $6\tan \dfrac{x}{2}-9{{\tan }^{2}}\dfrac{x}{2}=$
( a ) 0
( b ) 1
( c ) 3
( d ) 4

Mathematics
Trigonometric transformation formulae
If condition is given by \[A+B+C={{180}^{\circ }}\] then prove the following:
\[a.{{\sin }^{2}}A+{{\sin }^{2}}B-{{\sin }^{2}}C=2\sin A\sin B\cos C\]
\[b.{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=2\left( 1+\cos A\cos B\cos C \right)\]
Mathematics
Trigonometric transformation formulae
Find the derivative of ${{\cos }^{2}}x$, by using the first principle of derivatives.
Mathematics
Trigonometric transformation formulae
Using the trigonometric identity for sine of sum of angles, without expanding each term separately, prove that
\[sin\left( {{40}^{\circ }}+\theta \right)\cos \left( {{10}^{\circ }}+\theta \right)-\cos \left( {{40}^{\circ }}+\theta \right)\sin \left( {{10}^{\circ }}+\theta \right)=\dfrac{1}{2}\].

Mathematics
Trigonometric transformation formulae
Prove that:
$2\sin \left( \dfrac{5\pi }{12} \right)\sin \left( \dfrac{\pi }{12} \right)=\dfrac{1}{2}$
Mathematics
Trigonometric transformation formulae
Find the value of $\tan {22^0}30'$.
Mathematics
Trigonometric transformation formulae
What is the value of $cos\left( 36{}^\circ -A \right)cos\left( 36+A \right)+\cos \left( 54{}^\circ +A \right)\cos \left( 54{}^\circ -A \right)$ .
Mathematics
Trigonometric transformation formulae
Prove the trigonometric identity: $\dfrac{\cos {{11}^{\circ }}-\sin {{11}^{\circ }}}{\cos {{11}^{\circ }}+\sin {{11}^{\circ }}}=\cot {{56}^{\circ }}$

Mathematics
Trigonometric transformation formulae
Solve $1 + {\sin ^3}x + {\cos ^3}x - \frac{3}{2}\sin 2x = 0$
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